William M Boothby - Collected Quotes

"Apart from its own intrinsic interest, a knowledge of differentiable manifolds has become useful-ven mandatory-in an ever-increasing number of areas of mathematics and of its applications. This is not too surprising, since differentiable manifolds are the underlying, if unacknowledged, objects of study in much of advanced calculus and analysis. Indeed, such topics as line and surface integrals, divergence and curl of vector fields, and Stokes's and Green's theorems find their most natural setting in manifold theory. But however natural the leap from calculus on domains of Euclidean space to calculus on manifolds may be to those who have made it, it is not at all easy for most students." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"Curves and surfaces in Euclidean space were studied since the earliest days of geometry and, after they were invented, both analytic geometry and calculus were systematically used in these studies. However, the discoveries of Gauss, announced in 1827, profoundly altered the course of differential geometry and pointed the way to the concept of abstract difierentiable manifolds-the underlying spaces of every geometry and of other important mathematical theories as well. In his celebrated 'Theorema Egregium' Gauss showed that there is a measure of curvature of a surface (now called the Gaussian curvature) which depends only on one‘s ability to measure the lengths of curves on the surface. This means that this curvature is unchanged by alterations of shape of the surface which leave arclength unchanged. [...] This discovery of an 'inner' geometry, independent of the shape of the surface in E3, led very naturally toward the invention of abstract surfaces (2-manifolds) on which a measure of arclength is (somehow) provided." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"One of the most basic ideas used in the study of mappings from one space to another is that of homotopy. Two mappings are said to be homotopic if one can be 'deformed' into the other through a one-parameter family of mappings between the same spaces. Sometimes further conditions are imposed on the family of mappings [...]" (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"Our interest is in pointing out that new surfaces, that is, 2-manifolds, can be formed by fastening together manifolds with boundary along their boundaries, that is, by identifying points of various boundary components by a homeomorphism, assuming of course the necessary condition that such components are homeomorphic." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"There is, however, a unique feature of the tangent spaces of Euclidean space which is not shared by the tangent spaces at points of manifolds; the tangent spaces at any two points of Euclidean space are naturally isomorphic, that is, there is an isomorphism determined in some unique fashion by the geometry of the space - not chosen by us." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"A manifold M of dimension n, or n-manifold, is a topological space with the following properties: (i) M is Hausdorff, (ii) M is locally Euclidean of dimension n, and (iii) M has a countable basis of open sets." (William M Boothby, "An introduction to differentiable manifolds and Riemannian geometry" 2nd Ed., 1986)

On Manifolds (1975-1999)

"Apart from its own intrinsic interest, a knowledge of differentiable manifolds has become useful - even mandatory - in an ever-increasing number of areas of mathematics and of its applications. This is not too surprising, since differentiable manifolds are the underlying, if unacknowledged, objects of study in much of advanced calculus and analysis. Indeed, such topics as line and surface integrals, divergence and curl of vector fields, and Stokes's and Green's theorems find their most natural setting in manifold theory. But however natural the leap from calculus on domains of Euclidean space to calculus on manifolds may be to those who have made it, it is not at all easy for most students." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"Our interest is in pointing out that new surfaces, that is, 2-manifolds, can be formed by fastening together manifolds with boundary along their boundaries, that is, by identifying points of various boundary components by a homeomorphism, assuming of course the necessary condition that such components are homeomorphic." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"There is, however, a unique feature of the tangent spaces of Euclidean space which is not shared by the tangent spaces at points of manifolds; the tangent spaces at any two points of Euclidean space are naturally isomorphic, that is, there is an isomorphism determined in some unique fashion by the geometry of the space - not chosen by us." (William M Boothby, "Riemannian geometry: An introduction to differentiable Manifolds and Riemannian Geometry", 1975)

"A manifold M of dimension n, or n-manifold, is a topological space with the following properties: (i) M is Hausdorff, (ii) M is locally Euclidean of dimension n, and (iii) M has a countable basis of open sets." (William M Boothby, "An introduction to differentiable manifolds and Riemannian geometry" 2nd Ed., 1986)

"An attractor that consists of an infinite number of curves, surfaces, or higher-dimensional manifolds - generalizations of surfaces to multidimensional space - often occurring in parallel sets, with a gap between any two members of the set, is called a strange attractor." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Determination of transition functions makes it possible to restore the whole manifold if individual charts and coordinate maps are already given. Glueing functions may belong to different functional classes, which makes it possible to specify within a certain class of topological manifolds more narrow classes of smooth, analytic, etc. manifolds." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"Roughly speaking, manifolds are geometrical objects obtained by glueing open discs" (balls) like a papier-mache is glued of small paper scraps. To this end, one first prepares a clay or plastecine figure which is then covered with several sheets of paper scraps glued onto one another. After the plasticine is removed, there remains a two-dimensional surface." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Algebraic topology studies properties of a narrower class of spaces, - basically the classical objects of mathematics: spaces given by systems of algebraic and functional equations, surfaces lying in Euclidean space, and other sets which in mathematics are called manifolds. Examining the narrower class of spaces permits deeper penetration into their structure. (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

On Manifolds (1875-1899)

"With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it ‘mind-mass’. All thinking is, accordingly, formation of new mind masses." (Bernhard Riemann, "Gesammelte Mathematische Werke", 1876)

"If two well-defined manifolds M and N can be coordinated with each other univocally and completely, element by element (which, if possible in one way, can always happen in many others), we shall employ in the sequel the expression, that those manifolds have the same power or, also, that they are equivalent." (Georg Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre", 1878)

"The analysis of Nature into its individual parts, the grouping of the different natural processes and natural objects in definite classes, the study of the internal anatomy of organic bodies in their manifold forms - these were the fundamental conditions of the gigantic strides in our knowledge of Nature which have been made during the last four hundred years. But this method of investigation has also left us as a legacy the habit of observing natural objects and natural processes in their isolation, detached from the whole vast interconnection of things; and therefore not in their motion, but in their repose; not as essentially changing, but fixed constants; not in their life, but in their death." (Friedrich Engels, "Herr Eugen Dühring's Revolution in Science", 1878)

"I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined, when on the basis of its definition and as a consequence of the logical principle of excluded middle it must be regarded as internally determined, both whether an object pertaining to the same conceptual sphere belongs or not as an element to the manifold, and whether two objects belonging to the set are equal to each other or not, despite formal differences in the ways of determination." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic form or idea." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"The foregoing account of my researches in the theory of manifolds has reached a point where further progress depends on extending the concept of true integral number beyond the previous boundaries; this extension lies in a direction which, to my knowledge, no one has yet attempted to explore. My dependence on this extension of number concept is so great, that without it I should be unable to take freely the smallest step further in the theory of sets." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"Instead of the points of a line, plane, space, or any manifold under investigation, we may use instead any figure contained within the manifold: a group of points, curve, surface, etc. As there is nothing at all determined at the outset about the number of arbitrary parameters upon which these figures should depend, the number of dimensions of the line, plane, space, etc. is likewise arbitrary and depends only on the choice of space element. But so long as we base our geometrical investigation on the same group of transformations, the geometrical content remains unchanged. That is, every theorem resulting from one choice of space element will also be a theorem under any other choice; only the arrangement and correlation of the theorems will be changed. The essential thing is thus the group of transformations; the number of dimensions to be assigned to a manifold is only of secondary importance." (Felix Klein, "A comparative review of recent researches in geometry", Bulletin of the American Mathematoical Society 2(10), 1893)

On Manifolds (2000-2009)

"The Smale's horseshoe is the classical example of a structurally stable chaotic system: Its dynamical properties do not change under small perturbations, such as changes in control parameters. This is due to the horseshoe map being hyperbolic (i.e., the stable and unstable manifolds are transverse at each point of the invariant set)." (Robert Gilmore & Marc Lefranc, "TheTopologyof Chaos: Alice in Stretch and Squeezeland", 2002)

"By definition, a Kähler manifold is one with a complex structure (this means in particular that the coordinates changes are holomorphic for the complex coordinates) together with a Riemannian metric which has with this complex structure the best possible link, namely that multiplication of tangent vectors by unit complex numbers preserves the metric, but moreover the complex structure is invariant under parallel transport. This is equivalent to the condition that the holonomy group be included in the unitary group, hence equivalent also to ask for the existence of a 2-form of maximal rank and of zero covariant derivative."(Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"One reason for the importance of Riemannian manifolds is that they are generalizations of Euclidean geometry - general enough but not too general. They are still close enough to Euclidean geometry to have a Laplace operator. This is the key to quantum mechanics, heat and waves. The various generalizations of Riemannian manifold [...] do not have a simple natural unambiguous choice of such an operator. [...] Another reason for the prominence of Riemannian manifolds is that the maximal compact subgroup of the general linear group is the orthogonal group. So the least restriction we can make on any geometric structure so that it 'rigidifies' always adds a Riemannian geometry. Moreover, any geometric structure will always permit such a 'rigidification'. [...] Similarly, if we were to pick out a submanifold of the tangent bundle of some manifold, distinguishing tangent vectors, in such a manner that in each tangent space, any two lines could be brought to one another, or any two planes, etc., then the maximal symmetry group we could come up with in a single tangent space which was not the whole general linear group would be the orthogonal group of a Riemannian metric. So Riemannian geometry is the 'least' structure, or most symmetrical one, we can pick, at first order." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Riemann forged two simultaneous innovations: first, he defined (not too rigorously) a differentiable manifold to be a set of any dimension n, where one can perform differential calculus, change coordinates, etc. In particular, one has differentiable curves, tangent vectors (velocities) of those curves, and a tangent space at each point (i.e. all possible velocities of any curves through that point). Then he asked that a geometry on a manifold be simply an arbitrary positive definite quadratic form on each of those tangent spaces, thought of as the analogue of Gauß’s first fundamental form. One could use the same expression to define length of curves, look for shortest curves, etc." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"We have implicitly assumed that there are infinitely many periodic geodesics; [...] this is an open question. There is also the notion of geometrically different geodesics: turning more than once along a given periodic geodesic is not considered a different geodesic by a geometer, even if it might be in some sense different for a mechanics expert. The field of geodesic dynamics is dramatically different from that of spectrum geometry. The basic reason is that eigenfunctions are the critical points of the Dirichlet quotient on the (infinite dimensional) vector space of functions on the manifold. Periodic geodesics are the critical points of the length function on the space of all closed curves of the manifold. Sadly enough, this is not a vector space but an infinite dimensional manifold: one cannot play linear algebra with periodic geodesics." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Why should a geometer, whose principal concern is in measurements of distance, desire to engage in analysis on a Riemannian manifold? For example, pondering the Laplacian, its eigenvalues and eigenfunctions? Here are some reasons, chosen from among many others. We note also here that the existence of a canonical elliptic differential operator on any Riemannian manifold, one which is moreover easy to define and manipulate, is one of the motivations to consider Riemannian geometry as a basic field of investigation. [...] Riemannian geometry is by its very essence differential, working on manifolds with a differentiable structure. This automatically leads to analysis. It is interesting to note here that, historically, many great contributions to the field of Riemannian geometry came from analysts." (Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003)

"Manifolds are a type of topological spaces we are interested in. They correspond well to the spaces we are most familiar with, the Euclidean spaces. Intuitively, a manifold is a topological space that locally looks like Rn. In other words, each point admits a coordinate system, consisting of coordinate functions on the points of the neighborhood, determining the topology of the neighborhood." (Afra J Zomorodian, "Topology for Computing", 2005)

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005)

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Roughly speaking, a manifold is essentially a space that is locally similar to the Euclidean space. This resemblance permits differentiation to be defined. On a manifold, we do not distinguish between two different local coordinate systems. Thus, the concepts considered are just those independent of the coordinates chosen. This makes more sense if we consider the situation from the physics point of view. In this interpretation, the systems of coordinates are systems of reference." (Ovidiu Calin & Der-Chen Chang,  "Geometric Mechanics on Riemannian Manifolds : Applications to partial differential equations", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"A manifold is an abstract mathematical space, which locally (i.e., in a close–up view) resembles the spaces described by Euclidean geometry, but which globally (i.e., when viewed as a whole) may have a more complicated structure." (Vladimir G Ivancevic & Tijana T Ivancevic, "Applied Differential Geometry: A Modern Introduction", 2007)

"The idea of Morse theory is that the topology/geometry of a manifold can be understood by examining the smooth functions (and their singularities) on that manifold." (Steven G Krantz, "The Proof is in the Pudding", 2007)

"The most familiar manifold, however, is the space-time manifold, which has 4 dimensions. It is described by a time coordinate and three spatial coordinates. In addition to being a differentiable manifold, space-time has much more additional structure. It is at the level of this additional structure [...] that the space-time of Newtonian physics differs from the space-time of special relativity and from the space-times of Einstein’s theory of gravity (also called general relativity)." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

On Manifolds (1925-1949)

"An 'empty world', i. e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy." (Howard P Robertson, "On Relativistic Cosmology", 1928)

"Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but 'logical relations' or 'artificial manifolds'. They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"We must [...] maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time. [...] Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue… Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time. [...] the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"If a continuum, say, a two-dimensional closed manifold, a surface, is to be the subject of mathematícal investigation, t hen we must think of it as being subdivided into finitely many 'elementary pieces' whose topological nature is that of a circular disk. TP,ese pieces are further fragmented by repeated subdivision in accordance with a fixed scheme, and thus a particular spot in the continuum is ever more precisely intercepted by an infinite sequence 

of nested fragments that arise in t he course of successive subdivisions. In the onedimensional case, the repeated "normal subdivision" of an elementary segment is its bipartition. In the two-dirnensional case, each edge is first bipartitioned, then each piece of surface is divided into triangles by means of Unes in the surface that lead from a.n arbitrary center to the ( old and new) vertices." (Hermann Weyl, "Levels of Infinity", 1930)

"The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]" (Hermann Weyl, "Mathematics and Logic", 1946)

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

On Manifolds (1950-1974)

"A regular curve on a Riemannian manifold is a curve with a continuously turning nontrivial tangent vector. A regular homotopy is a homotopy which at every stage is a regular curve, keeps end points and directions fixed and such that the tangent vector moves continuously with the homotopy. A regular curve is closed if its initial point and tangent  coincides with its end point and tangent." (Steven Smale,"Regular Curves on Riemann Manifolds", 1956)

"The stable manifolds of the critical points of a nice function can be thought of as the cells of a complex while the unstable manifolds are the dual cells. This structure has the advantage over previous structures that both the cells and the duals are differentiably imbedded in M. We believe that nice functions will replace much of the use of С triangulations and combinatorial methods in differential topology." (Steven Smale, "The generalized Poincare conjecture in higher dimensions", Bull. Amer. Math. Soc. 66, 1960)

"Certainly, the problems of combinatorial manifolds and the relationships between combinatorial and differentiable manifolds are legitimate problems in their own right. An example is the questionof existence and uniqueness of differentiable structures on a combinatorial manifold. However, we don't believe such problems are the goal of differential topology itself. This view seems justified by the fact that today one can substantially develop differential topology most simply without any reference to the combinatorial manifolds." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"[...] it is clear that differential geometry, analysis and physics prompted the early development of differential topology (it is this that explains our admitted bias toward differential topology, that it lies close to the main stream of  mathematics). On the other hand, the combinatorial approach to manifolds was started because it was believed that these means would afford a useful attack on the differentiable case." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"We consider differential topology to be the study of differentiable manifolds and differentiable maps. Then, naturally, manifolds are considered equivalent if they are diffeomorphic, i.e., there exists a differentiable map from one to the other with a differentiable inverse." (Steven Smale, "A survey of some recent developments in differential topology", 1961)

"Historically speaking, topology has followed two principal lines of development. In homology theory, dimension theory, and the study of manifolds, the basic motivation appears to have come from geometry. In these fields, topological spaces are looked upon as generalized geometric configurations, and the emphasis is placed on the structure of the spaces themselves. In the other direction, the main stimulus has been analysis. Continuous functions are the chief objects of interest here, and topological spaces are regarded primarily as carriers of such functions and as domains over which they can be integrated. These ideas lead naturally into the theory of Banach and Hilbert spaces and Banach algebras, the modern theory of integration, and abstract harmonic analysis on locally compact groups." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"A manifold can be given by specifying the coordinate ranges of an atlas, the images in those coordinate ranges of the overlapping parts of the coordinate domains, and the coordinate transformations for each of those overlapping domains. When a manifold is specified in this way, a rather tricky condition on the specifications is needed to give the Hausdorff property, but otherwise the topology can be defined completely by simply requiring the coordinate maps to be homeomorphisms." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"In the definition of a coordinate system we have required that the coordinate neighborhood and the range in Rd be open sets. This is contrary to popular usage, or at least more specific than the usage of curvilinear coordinates in advanced calculus. For example, spherical coordinates are used even along points of the z axis where they are not even 1-1. The reasons for the restriction to open sets are that it forces a uniformity in the local structure which simplifies analysis on a manifold (there are no 'edge points') and, even if local uniformity were forced in some other way, it avoids the problem of. spelling out what we mean by differentiability at boundary points of the coordinate neighborhood; that is, one-sided derivatives need not be mentioned. On the other hand, in applications, boundary value problems frequently arise, the setting for which is a manifold with boundary. These spaces are more general than manifolds and the extra generality arises from allowing a boundary manifold of one dimension less. The points of the boundary manifold have a coordinate neighborhood in the boundary manifold which is attached to a coordinate neighborhood of the interior in much the same way as a face of a cube is attached to the interior. Just as the study of boundary value problems is more difficult than the study of spatial problems, the study of manifolds with boundary is more difficult than that of mere manifolds, so we shall limit ourselves to the latter." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"My definition of global analysis is simply the study of differential equations, both ordinary and partial, on manifolds and vector space bundles. Thus one might consider global analysis as differential equations from a global, or topological point of view." (Steven Smale, "What is global analysis?", American Mathematical Monthly Vol. 76 (1), 1969)

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves." (Shiing-Shen Chern, "Differential geometry, its past and its future", 1970)

"A fairly general procedure for mathematical study of a physical system with explication of the space of states of that system. Now this space of states could reasonably be one of a number of mathematical objects. However, in my mind, a principal candidate For the state space should be a differentiable manifold; and in case the has a finite number of degrees of freedom, then this will be a finite dimensional manifold. Usually associated with physical is the notion of how a state progresses in time. The corresponding object is a dynamical system or a first order ordinary differential equation on the manifold of states." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

"Related is the idea of structural stability and certain variations. This kind of is a property of a dynamical system itself (not of a or orbit) and asserts that nearby dynamical systems have the same structure. The 'same structure' can be defined in several interesting ways, but the basic idea is that two dynamical systems have 'the same structure' if they have the same gross behavior, or the same qualitative behavior. For example, the original definition of 'same structure' of two dynamical systems was that there was an orbit preserving continuous transformation between them. This yields the definition of structural stability proper. It is a recent theorem that every compact manifold admits structurally stable systems, and almost all gradient dynamical systems are structurally stable. But while there exists a rich set of structurally stable systems, there are also important examples which are not stable, and have good but weaker stability properties." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

On Manifolds (1850-1874)

"If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forward or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded threefold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesies starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1854)

"Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

"If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

"Magnitude-notions are only possible where there is an antecedent general notion which admits of different specialisations. According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements, of the manifoldness." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

On Manifolds (1900-1924)

"The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space." (Paul Carus, Science Vol. 18, 1903)

"A mathematical theorem and its demonstration are prose. But if the mathematician is overwhelmed with the grandeur and wondrous harmony of geometrical forms, of the importance and universal application of mathematical maxims, or, of the mysterious simplicity of its manifold laws which are so self-evident and plain and at the same time so complicated and profound, he is touched by the poetry of his science; and if he but understands how to give expression to his feelings, the mathematician turns poet, drawing inspiration from the most abstract domain of scientific thought." (Paul Carus, „Friedrich Schiller: A Sketch of His Life and an Appreciation of His Poetry", 1905)

"In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system." (Charles P Steinmetz, [New York Times interview] 1911)

"That branch of mathematics which deals with the continuity properties of two- (and more) dimensional manifolds is called analysis situs or topology. […] Two manifolds must be regarded as equivalent in the topological sense if they can be mapped point for point in a reversibly neighborhood-true (topological) fashion on each other." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"In her manifold opportunities Nature has thus helped man to polish the mirror of [man’s] mind, and the process continues. Nature still supplies us with abundance of brain-stretching theoretical puzzles and we eagerly tackle them; there are more worlds to conquer and we do not let the sword sleep in our hand; but how does it stand with feeling? Nature is beautiful, gladdening, awesome, mysterious, wonderful, as ever, but do we feel it as our forefathers did?" (Sir John A Thomson, "The System of Animate Nature", 1920)

On Manifolds (2010-)

"A topological manifold of dimension k is a Hausdorff topological space M with a countable base such that for all x ∈ M, there exists an open neighborhood of x that is homeomorphic to an open set of Rk." (Stephen Lovett, "Differential Geometry of Manifolds", 2010)

"Roughly speaking, a manifold is a set whose points can be labeled by coordinates." (Gerardo F. Torres del Castillo, "Differentiable Manifolds: A Theoretical Physics Approach", 2010)

"One of the most important skills you will need to acquire in order to use manifold theory effectively is an ability to switch back and forth easily between invariant descriptions and their coordinate counterparts." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)

"The fact that manifolds do not come with any predetermined choice of coordinates is both a blessing and a curse. The flexibility to choose coordinates more or less arbitrarily can be a big advantage in approaching problems in manifold theory, because the coordinates can often be chosen to simplify some aspect of the problem at hand. But we pay for this flexibility by being obliged to ensure that any objects we wish to define globally on a manifold are not dependent on a particular choice of coordinates. There are generally two ways of doing this: either by writing down a coordinate-dependent definition and then proving that the definition gives the same results in any coordinate chart, or by writing down a definition that is manifestly coordinate-independent (often called an invariant definition)." (John M Lee, "Introduction to Smooth Manifolds" 2nd Ed., 2013)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"You can very generally think of a manifold as a space which is locally Euclidian - that means that if you look closely enough at one small part of a manifold then it basically looks like Rn for some n." (Jon P Fortney, "A Visual Introduction to Differential Forms and Calculus on Manifolds", 2018)

On Manifolds: Definitions

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A manifold M of dimension n, or n-manifold, is a topological space with the following properties: (i) M is Hausdorff, (ii) M is locally Euclidean of dimension n, and (iii) M has a countable basis of open sets." (William M Boothby, "An introduction to differentiable manifolds and Riemannian geometry" 2nd Ed., 1986)

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"A manifold Mn of dimension n is a Hausdorff topological space such that each point P of Mn has a neighborhood Ω homeomorphic to Rn (or equivalently to an open set of Rn." (Thierry Aubin, "A Course in Differential Geometry", 2000)

"Manifolds are a type of topological spaces we are interested in. They correspond well to the spaces we are most familiar with, the Euclidean spaces. Intuitively, a manifold is a topological space that locally looks like Rn. In other words, each point admits a coordinate system, consisting of coordinate functions on the points of the neighborhood, determining the topology of the neighborhood." (Afra J Zomorodian, "Topology for Computing", 2005)

"Roughly speaking, a manifold is essentially a space that is locally similar to the Euclidean space. This resemblance permits differentiation to be defined. On a manifold, we do not distinguish between two different local coordinate systems. Thus, the concepts considered are just those independent of the coordinates chosen. This makes more sense if we consider the situation from the physics point of view. In this interpretation, the systems of coordinates are systems of reference." (Ovidiu Calin & Der-Chen Chang,  "Geometric Mechanics on Riemannian Manifolds : Applications to partial differential equations", 2005)

"A manifold is an abstract mathematical space, which locally (i.e., in a close–up view) resembles the spaces described by Euclidean geometry, but which globally (i.e., when viewed as a whole) may have a more complicated structure." (Vladimir G Ivancevic & Tijana T Ivancevic, "Applied Differential Geometry: A Modern Introduction", 2007)

"A topological manifold of dimension k is a Hausdorff topological space M with a countable base such that for all x ∈ M, there exists an open neighborhood of x that is homeomorphic to an open set of Rk." (Stephen Lovett, "Differential Geometry of Manifolds", 2010)

"Roughly speaking, a manifold is a set whose points can be labeled by coordinates." (Gerardo F. Torres del Castillo, "Differentiable Manifolds: A Theoretical Physics Approach", 2010)

"You can very generally think of a manifold as a space which is locally Euclidian - that means that if you look closely enough at one small part of a manifold then it basically looks like Rn for some n." (Jon P Fortney, "A Visual Introduction to Differential Forms and Calculus on Manifolds", 2018)

On Differentiability I

"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] differentiation is performed by focusing on the behavior of a function near one point. A quantity obtained in this manner is essentially a local quantity. Is it possible that such local quantities can show us something very basic about global properties such as smoothness? Does there exist a place in mathematics which would enable us to study the relationship between local and global quantities?" (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"To consider differentiable functions, we must introduce a coordinate system on the plane and thereby to concentrate on the world of numbers.[...] a continuous function defined on a plane can be differentiable or nondifferentiable depending on the choice of coordinates. [...] the choice of coordinates on the plane determines which functions among the continuous functions should be selected as differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"I turn away with fright and horror from the lamentable evil of functions which do not have derivatives." (Charles Hermite, [letter to Thomas J Stieltjes])

Richard L Bishop - Collected Quotes

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A manifold can be given by specifying the coordinate ranges of an atlas, the images in those coordinate ranges of the overlapping parts of the coordinate domains, and the coordinate transformations for each of those overlapping domains. When a manifold is specified in this way, a rather tricky condition on the specifications is needed to give the Hausdorff property, but otherwise the topology can be defined completely by simply requiring the coordinate maps to be homeomorphisms." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"A set is countable if it is either finite or its members can be arranged in an infinite sequence; or, what is the same, there is a 1-1 map from the set into the positive integers."(Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"An initial study of tensor analysis can. almost ignore the topological aspects since the topological assumptions are either very natural (continuity, the Hausdorff property) or highly technical (separability, paracompactness). However, a deeper analysis of many of the existence problems encountered in tensor analysis requires assumption of some of the more difficult-to-use topological properties, such as compactness and paracompactness."  (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"In the definition of a coordinate system we have required that the coordinate neighborhood and the range in Rd be open sets. This is contrary to popular usage, or at least more specific than the usage of curvilinear coordinates in advanced calculus. For example, spherical coordinates are used even along points of the z axis where they are not even 1-1. The reasons for the restriction to open sets are that it forces a uniformity in the local structure which simplifies analysis on a manifold (there are no 'edge points') and, even if local uniformity were forced in some other way, it avoids the problem of. spelling out what we mean by differentiability at boundary points of the coordinate neighborhood; that is, one-sided derivatives need not be mentioned. On the other hand, in applications, boundary value problems frequently arise, the setting for which is a manifold with boundary. These spaces are more general than manifolds and the extra generality arises from allowing a boundary manifold of one dimension less. The points of the boundary manifold have a coordinate neighborhood in the boundary manifold which is attached to a coordinate neighborhood of the interior in much the same way as a face of a cube is attached to the interior. Just as the study of boundary value problems is more difficult than the study of spatial problems, the study of manifolds with boundary is more difficult than that of mere manifolds, so we shall limit ourselves to the latter." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"Set theory is concerned with abstract objects and their relation to various collections which contain them. We do not define what a set is but accept it as a primitive notion. We gain an intuitive feeling for the meaning of sets and, consequently, an idea of their usage from merely listing some of the synonyms: class, collection, conglomeration, bunch, aggregate. Similarly, the notion of an object is primitive, with synonyms element and point. Finally, the relation between elements and sets, the idea of an element being in a set, is primitive." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

On Manifolds II (Geometry II)

"In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent; the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded threefold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is forever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesies starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

"In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system." (Charles P Steinmetz, [New York Times interview] 1911)

"That branch of mathematics which deals with the continuity properties of two- (and more) dimensional manifolds is called analysis situs or topology. […] Two manifolds must be regarded as equivalent in the topological sense if they can be mapped point for point in a reversibly neighborhood-true (topological) fashion on each other." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"The power of differential calculus is that it linearizes all problems by going back to the 'infinitesimally small', but this process can be used only on smooth manifolds. Thus our distinction between the two senses of rotation on a smooth manifold rests on the fact that a continuously differentiable coordinate transformation leaving the origin fixed can be approximated by a linear transformation at О and one separates the (nondegenerate) homogeneous linear transformations into positive and negative according to the sign of their determinants. Also the invariance of the dimension for a smooth manifold follows simply from the fact that a linear substitution which has an inverse preserves the number of variables." (Hermann Weyl, "The Concept of a Riemann Surface", 1913)

"In her manifold opportunities Nature has thus helped man to polish the mirror of [man’s] mind, and the process continues. Nature still supplies us with abundance of brain-stretching theoretical puzzles and we eagerly tackle them; there are more worlds to conquer and we do not let the sword sleep in our hand; but how does it stand with feeling? Nature is beautiful, gladdening, awesome, mysterious, wonderful, as ever, but do we feel it as our forefathers did?" (Sir John A Thomson, "The System of Animate Nature", 1920)

"An 'empty world', i. e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy." (Howard P Robertson, "On Relativistic Cosmology", 1928)

"Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i. e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but 'logical relations' or 'artificial manifolds'. They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"We must [...] maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

On Manifolds V (Geometry III)

"Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time. [...] Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue… Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time. [...] the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold." (Hans Reichenbach, "The Philosophy of Space and Time", 1928)

"The sequence of numbers which grows beyond any stage already reached by passing to the next number is a manifold of possibilities open towards infinity, it remains forever in the status of creation, but is not a closed realm of things existing in themselves. That we blindly converted one into the other is the true source of our difficulties […]" (Hermann Weyl, "Mathematics and Logic", 1946)

"The first attempts to consider the behavior of so-called 'random neural nets' in a systematic way have led to a series of problems concerned with relations between the 'structure' and the 'function' of such nets. The 'structure' of a random net is not a clearly defined topological manifold such as could be used to describe a circuit with explicitly given connections. In a random neural net, one does not speak of 'this' neuron synapsing on 'that' one, but rather in terms of tendencies and probabilities associated with points or regions in the net." (Anatol Rapoport, "Cycle distributions in random nets", The Bulletin of Mathematical Biophysics 10(3), 1948)

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves." (Shiing-Shen Chern, "Differential geometry, its past and its future", 1970)

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)

"Therefore one has taken everywhere the opposite road, and each time one encounters manifolds of several dimensions in geometry, as in the doctrine of definite integrals in the theory of imaginary quantities, one takes spatial intuition as an aid. It is well known how one gets thus a real overview over the subject and how only thus are precisely the essential points emphasized." (Bernhard Riemann)

On Manifolds I (Geometry I)

"If in the case of a notion whose specialisations form a continuous manifoldness, one passes from a certain specialisation in a definite way to another, the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forward or backwards. If one now supposes that this manifoldness in its turn passes over into another entirely different, and again in a definite way, namely so that each point passes over into a definite point of the other, then all the specialisations so obtained form a doubly extended manifoldness. In a similar manner one obtains a triply extended manifoldness, if one imagines a doubly extended one passing over in a definite way to another entirely different; and it is easy to see how this construction may be continued. If one regards the variable object instead of the determinable notion of it, this construction may be described as a composition of a variability of n + 1 dimensions out of a variability of n dimensions and a variability of one dimension." (Bernhard Riemann, "On the hypotheses which lie at the foundation of geometry", 1854)

"Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

"Magnitude-notions are only possible where there is an antecedent general notion which admits of different specialisations. According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements, of the manifoldness." (Bernhard Riemann, "On the Hypotheses which lie at the Bases of Geometry", 1873)

"With every simple act of thinking, something permanent, substantial, enters our soul. This substantial somewhat appears to us as a unit but (in so far as it is the expression of something extended in space and time) it seems to contain an inner manifoldness; I therefore name it ‘mind-mass’. All thinking is, accordingly, formation of new mind masses." (Bernhard Riemann, "Gesammelte Mathematische Werke", 1876)

"If two well-defined manifolds M and N can be coordinated with each other univocally and completely, element by element (which, if possible in one way, can always happen in many others), we shall employ in the sequel the expression, that those manifolds have the same power or, also, that they are equivalent." (Georg Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre", 1878)

"I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined, when on the basis of its definition and as a consequence of the logical principle of excluded middle it must be regarded as internally determined, both whether an object pertaining to the same conceptual sphere belongs or not as an element to the manifold, and whether two objects belonging to the set are equal to each other or not, despite formal differences in the ways of determination." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic form or idea." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"The truth is that other systems of geometry are possible, yet after all, these other systems are not spaces but other methods of space measurements. There is one space only, though we may conceive of many different manifolds, which are contrivances or ideal constructions invented for the purpose of determining space." (Paul Carus, Science Vol. 18, 1903)

"A mathematical theorem and its demonstration are prose. But if the mathematician is overwhelmed with the grandeur and wondrous harmony of geometrical forms, of the importance and universal application of mathematical maxims, or, of the mysterious simplicity of its manifold laws which are so self-evident and plain and at the same time so complicated and profound, he is touched by the poetry of his science; and if he but understands how to give expression to his feelings, the mathematician turns poet, drawing inspiration from the most abstract domain of scientific thought." (Paul Carus, „Friedrich Schiller: A Sketch of His Life and an Appreciation of His Poetry", 1905)

"Roughly speaking, a manifold is essentially a space that is locally similar to the Euclidean space. This resemblance permits differentiation to be defined. On a manifold, we do not distinguish between two different local coordinate systems. Thus, the concepts considered are just those independent of the coordinates chosen. This makes more sense if we consider the situation from the physics point of view. In this interpretation, the systems of coordinates are systems of reference." (Ovidiu Calin & Der-Chen Chang,  "Geometric Mechanics on Riemannian Manifolds : Applications to partial differential equations", 2005)

On Manifolds IV (Trivia II)

"The object of pure mathematics is those relations which may be conceptually established among any conceived elements whatsoever by assuming them contained in some ordered manifold; the law of order of this manifold must be subject to our choice; the latter is the case in both of the only conceivable kinds of manifolds, in the discrete as well as in the continuous." (Erwin Papperitz, "Über das System der rein mathematischen Wissenschaften", 1910)

"As systematic inquiry into the natural facts was begun it was at once found that the accepted ideas of variation were unfounded. Variation was seen very frequently to be a definite and specific phenomenon, affecting different forms of life in different ways, but in all its diversity showing manifold and often obvious indications of regularity." (William Bateson, "Problems in Genetics", 1913)

"The validity of demonstrably wrong law cannot conceivably be justified. However, any answer to the question of the purpose of law other than by enumerating the manifold partisan views about it has proved impossible - and it is precisely on that impossibility of any natural law, and on that alone, that the validity of positive law may be founded. At this point relativism, so far only the method of our approach, enters our system as a structural element." (Gustav Radbruch, "Rechtsphilosophie", 1932)

"We know, since the theory of relativity at least, that empirical sciences are to some degree free in defining dynamical concepts or even in assuming laws, and that only a system as a whole which includes concepts, coordinating definitions, and laws can be said to be either true or false, to be adequate or inadequate to empirical facts. This 'freedom', however, is a somewhat doubtful gift. The manifold of possibilities implies uncertainty, and such uncertainty can become rather painful in a science as young as psychology, where nearly all concepts are open and unsettled. As psychology approaches the state of a logically sound science, definitions cease to be an arbitrary matter. They become far-reaching decisions which presuppose the mastering of the conceptual problems but which have to be guided entirely by the objective facts." (Kurt Lewin, "Principles of topological psychology", 1936)

"The true physician cannot remain outside the manifold of the events he observes." (Alan Gregg, "Humanism and Science", Bulletin of the New York Academy of Sciences Vol. 17, 1941)

"The mystery that clings to numbers, the magic of numbers, may spring from this very fact, that the intellect, in the form of the number series, creates an infinite manifold of well-distinguished individuals. Even we enlightened scientists can still feel it, e.g., in the impenetrable law of the distribution of prime numbers." (Hermann Weyl, "Philosophy of Mathematics and Natural Science", 1949)

"[...] our purpose is to give a presentation of geometry [..[.] in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems. [...] beyond this, it is possible [...] to depict the geometric outline of the methods of investigation and proof, without [...] entering into the details [...] In this manner, geometry being as many-faceted as it is and being related to the most diverse branches of mathematics, we may even obtain a summarizing survey of mathematics as a whole, and a valid idea of the variety of problems and the wealth of ideas it contains. Thus a presentation of geometry in large brushstrokes [...] and based on the approach through visual intuition, should contribute to a more just appreciation of mathematics by a wider range of people than just the specialists." (David Hilbert, "Geometry and the Imagination", 1952)

"The historian's special contribution is the discovery of the manifold shapes of time. The aim of the historian, regardless of his specialty in erudition, is to portray time. He is committed to the detection and description of the shape of time." (George Kubler, "The Shape of Time", 1982)

"People are deeply imbedded in philosophical, i.e., grammatical confusions. And to free them presupposes pulling them out of the immensely manifold connections they are caught up in." (Ludwig Wittgenstein, "Philosophical Occasions 1912-1951", 1993)

"Direct experience is inherently too limited to form an adequate foundation either for theory or for application. At the best it produces an atmosphere that is of value in drying and hardening the structure of thought. The greater value of indirect experience lies in its greater variety and extent. History is universal experience, the experience not of another, but of many others under manifold conditions." (Basil L Hart, "Why Don't We Learn from History?", 2015)

On Manifolfd III (Trivia I)

"Philosophers conceive of the passions which harass us as vices into which men fall by their own fault, and, therefore, generally deride, bewail, or blame them, or execrate them, if they wish to seem unusually pious. And so they think they are doing something wonderful, and reaching the pinnacle of learning, when they are clever enough to bestow manifold praise on such human nature, as is nowhere to be found, and to make verbal attacks on that which, in fact, exists. For they conceive of men, not as they are, but as they themselves would like them to be. Whence it has come to pass that, instead of ethics, they have generally written satire, and that they have never conceived a theory of politics, which could be turned to use, but such as might be taken for a chimera, or might have been formed in Utopia, or in that golden age of the poets when, to be sure, there was least need of it. Accordingly, as in all sciences, which have a useful application, so especially in that of politics, theory is supposed to be at variance with practice; and no men are esteemed less fit to direct public affairs than theorists or philosophers." (Baruch Spinoza, "Political Treatise", 1677)

"All true metaphysics is taken from the essential nature of the thinking faculty itself, and therefore in nowise invented, since it is not borrowed from experience, but contains the pure operations of thought, that is, conceptions and principles à priori, which the manifold of empirical presentations first of all brings into legitimate connection, by which it can become empirical knowledge, i.e. experience. [...] mathematical physicists were thus quite unable to dispense with such metaphysical principles [...]" (Immanuel Kant, "Metaphysical Foundations of Natural Science", 1786)

"It is the principle of necessity towards which, as to their ultimate centre, all the ideas advanced in this essay immediately converge. In abstract theory the limits of this necessity are determined solely by considerations of man’s proper nature as a human being; but in the application we have to regard, in addition, the individuality of man as he actually exists. This principle of necessity should, I think, prescribe the grand fundamental rule to which every effort to act on human beings and their manifold relations should be invariably conformed. For it is the only thing which conducts to certain and unquestionable results. The consideration of the useful, which might be opposed to it, does not admit of any true and unswerving decision." (Wilhelm Von Humboldt, "The Limits of State Action", 1792)

"Before abstraction everything is one, but one like chaos; after abstraction everything is united again, but this union is a free binding of autonomous, self-determined beings. Out of a mob a society has developed, chaos has been transformed into a manifold world." (G P Friedrich F von Hardenberg [Novalis], "Blüthenstaub" [Fragment No. 95], 1798)

"Nature, in the manifold signification of the word - whether considered as the universality of all that is and ever will be - as the inner moving force of all phenomena, or as their mysterious prototype - reveals itself to the simple mind and feelings of man as something earthly, and closely allied to himself. "(Alexander von Humboldt, Cosmos: "A Sketch of a Physical Description of the Universe", 1845)

"Nothing protects us so surely from this wrong turning as inner wealth, the wealth of the mind, for the more eminent it becomes, the less room does it leave for boredom. The inexhaustible activity of ideas, their constantly renewed play with the manifold phenomena of the inner and outer worlds, the power and urge always to make different combinations of them, all these put the eminent mind, apart from moments of relaxation, quite beyond the reach of boredom." (Arthur Schopenhauer, "Parerga and Paralipomena", 1851)

"Manifold subsequent experience has led to a truer appreciation and a more moderate estimate of the importance of the dependence of one living being upon another." (Richard Owen, The Edinburgh Review, 1860)

"If we consider further the manifold relations of this mathematical theory to civil uses and the technical arts, we shall recognize completely the extent of its applications. It is evident that it includes an entire series of distinct phenomena, and that the study of it cannot be omitted without losing a notable part of the science of nature.
The principles of the theory are derived, as are those of rational mechanics, from a very small number of primary facts, the causes of which are not considered by geometers, but which they admit as the results of common observations confirmed by all experiment." (Joseph Fourier, "The Analytical Theory of Heat", 1878)

"The analysis of Nature into its individual parts, the grouping of the different natural processes and natural objects in definite classes, the study of the internal anatomy of organic bodies in their manifold forms—these were the fundamental conditions of the gigantic strides in our knowledge of Nature which have been made during the last four hundred years. But this method of investigation has also left us as a legacy the habit of observing natural objects and natural processes in their isolation, detached from the whole vast interconnection of things; and therefore not in their motion, but in their repose; not as essentially changing, but fixed constants; not in their life, but in their death." (Friedrich Engels, "Herr Eugen Dühring's Revolution in Science", 1878)

"A philosophy which emphasises the idea of the One, is generally a dualistic philosophy in which the conception of Second receives exaggerated attention: for this One (though of course involving the idea of First) is always the other of a manifold which is not one." (Charles S Peirce, "The Architecture of Theories", 1891)

Kenji Ueno - Collected Quotes

"[...] a manifold is a set M on which 'nearness' is introduced (a topological space), and this nearness can be described at each point in M by using coordinates. It also requires that in an overlapping region, where two coordinate systems intersect, the coordinate transformation is given by differentiable transition functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"An essential difference between continuity and differentiability is whether numbers are involved or not. The concept of continuity is characterized by the qualitative property that nearby objects are mapped to nearby objects. However, the concept of differentiation is obtained by using the ratio of infinitesimal increments. Therefore, we see that differentiability essentially involves numbers." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Analyticity is the property of a differentiable function y = f(x) that can be represented by the infinite series for all x near each point x0." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"By studying analytic functions using power series, the algebra of the Middle Ages was connected to infinite operations (various algebraic operations with infinite series). The relation of algebra with infinite operations was later merged with the newly developed differential and integral calculus. These developments gave impetus to early stages of the development of analysis. In a way, we can say that analyticity is the notion that first crossed the boundary from finite to infinite by passing from polynomials to infinite series. However, algebraic properties of polynomial functions still are strongly present in analytic functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[…] continuity appears when we try to mathematically express continuously changing phenomena, and differentiability is the result of expressing smoothly changing phenomena."  (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Continuous functions can change freely, while analytic functions are rigid. In this sense, we can say that continuous and analytic functions are antipodal." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Continuous functions can move freely. Graphs of continuous functions can freely branch off at any place, whereas analytic functions coinciding in some neighborhood of a point P cannot branch outside of this neighborhood. Because of this property, continuous functions can mathematically represent wildly changing wind inside a typhoon or a gentle breeze." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Differentiability of a function can be established by examining the behavior of the function in the immediate neighborhood of a single point a in its domain. Thus, all we need is coordinates in the vicinity of the point a. From this point of view, one might say that local coordinates have more essential qualities. However, if are not looking at individual surfaces, we cannot find a more general and universal notion than smoothness." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] differentiation is performed by focusing on the behavior of a function near one point. A quantity obtained in this manner is essentially a local quantity. Is it possible that such local quantities can show us something very basic about global properties such as smoothness? Does there exist a place in mathematics which would enable us to study the relationship between local and global quantities?" (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Even though there are no methods to represent each differentiable function by an 'equation', we can still investigate differentiable functions by various analytic methods. Because of this, we can say that differentiable functions have more mathematical reality than continuous functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Here we have the following question: 'Which concept is closer to the concept of differentiability -continuity or analyticity?' The answer depends upon the point of view. Our point of view is that continuity appears when we try to mathematically express continuously changing phenomena, and differentiability is the result of expressing smoothly changing phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If each change of a certain quantity results in a corresponding change of another quantity, we can say that there exists a functional relationship between those two quantities. Viewed in this manner, the idea of functions expands endlessly. The concept of functions is truly comprehensive, but while it is all encompassing, it is not fathomless; at least, not with respect to our current subject of manifolds. You might feel that linear functions or quadratic functions are far too specific and that you are sinking into the depths of the ocean called functions. However, you will be rescued from the ocean depths by understanding of the functions that are needed to describe manifolds. These functions are continuous, analytic, and differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] if we consider a topological space instead of a plane, then the question of whether the coordinates axes in that space are curved or straight becomes meaningless. The way we choose coordinate systems is related to the way we observe the property of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"If we know when a sequence approaches a point or, as we say, converges to a point, we can define a continuous mapping from one metric space to another by using the property that a converging sequence is mapped to the corresponding converging sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"In abstract mathematics, special attention is given to particular properties of numbers. Then those properties are taken in a very pure (and primitive) form. Those properties in pure form are then assigned to a given set. Therefore, by studying in details the internal mathematical structure of a set, we should be able to clarify the meaning of original properties of the objects. Likewise, in set theory, numbers disappear and only the concept of sets and characteristic properties of sets remain." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"In view of the developments of abstract mathematics, the first thing mathematicians studied was how to extract the property of 'nearness' from the set of numbers. If the property of nearness could be extracted using a few axioms, and if it was possible to associate the extracted property with a set, then the resulting set would provide an abstract scene to study 'nearness'." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Intuitively speaking, a visual representation associated with the concept of continuity is the property that a near object is sent to a corresponding near object, that is, a convergent sequence is sent to a corresponding convergent sequence." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"It is commonly said that the study of manifolds is, in general, the study of the generalization of the concept of surfaces. To some extent, this is true. However, defining it that way can lead to overshadowing by 'figures' such as geometrical surfaces." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] moving from the concept of continuity to differentiability, and then to analyticity, we are moving from weaker properties to stronger ones. Therefore, the relations between the corresponding properties of functions can be expressed as follows: {continuous functions} > {differentiate functions} > {analytic functions}." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Similarly to the graphs of continuous functions, graphs of differentiable (smooth) functions which coincide in a neighborhood of a point P can branch off outside of the neighborhood. Because of this property, differentiable functions can represent smoothly changing natural phenomena." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"The best example to indicate the rigidity of analytic functions is a soap film (with little viscosity) on a wire frame (think of a bubble blower). The soap film, which is created by the surface tension, stretches across the wire frame and is known to have analyticity. Therefore, if we try to change a certain region of the film by tapping it with a stick, then the film loses analyticity and will immediately brake." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"[...] the only characteristic property that continuous functions have is that near objects are sent to corresponding near objects, that is, a convergent sequence is mapped to the corresponding convergent sequence. It is reasonable to say that we cannot expect to extract from that property neither numerical consequences, nor a method to extensively study continuity. On the contrary, analytic functions can be represented by equations (precisely speaking, by infinite series). Compared to analytic functions, continuous functions, in general, are difficult to represent explicitly, although they exist as a concept." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"The property of smoothness includes the property of continuity. The notion of a topological space was born from the development of abstract algebra as a universal notion for the property of continuity." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"To consider differentiable functions, we must introduce a coordinate system on the plane and thereby to concentrate on the world of numbers.[...] a continuous function defined on a plane can be differentiable or nondifferentiable depending on the choice of coordinates. [...] the choice of coordinates on the plane determines which functions among the continuous functions should be selected as differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"To describe the property of smoothness, differentiable functions should be specified first. To do so, coordinates need to be introduced on the topological space. Those coordinates can be local coordinates such as the ones used by Gauss. Once coordinates are introduced around a point a in a topological space, differentiable functions near the point a are distinguished from the continuous functions in the region near a. If different coordinates are chosen, then a different set of differentiable functions is distinguished. In other words, the choice of local coordinates determines the notion of smoothness in a topological space." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"When the absolute concept of coordinate systems introduced by Descartes shifted to the relative concept of coordinate systems introduced by Gauss, a clear differences between continuity and differentiability emerged."  (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"When we study the concept of continuity by itself, numbers are not necessary as long as you are dealing with objects which have the property of 'nearness' . Therefore, if we can introduce the notion of 'nearness' detached from numbers from a purely abstract point of view, then we can discuss topics related to continuity based upon this notion. This approach enables us to become familiar with the science we call mathematics." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)