On Curvature II

"In order to regain in a rigorously defined function those properties that are analogous to those ascribed to an empirical curve with respect to slope and curvature (first and higher difference quotients), we need not only to require that the function is continuous and has a finite number of maxima and minima in a finite interval, but also assume explicitly that it has the first and a series of higher derivatives (as many as one will want to use)." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)

"Any region of space-time that has no gravitating mass in its vicinity is uncurved, so that the geodesics here are straight lines, which means that particles move in straight courses at uniform speeds (Newton's first law). But the world-lines of planets, comets and terrestrial projectiles are geodesics in a region of space-time which is curved by the proximity of the sun or earth. […] No force of gravitation is […] needed to impress curvature on world-lines; the curvature is inherent in the space […]" (James H Jeans," The Growth of Physical Science", 1947) 

"Space-time is curved in the neighborhood of material masses, but it is not clear whether the presence of matter causes the curvature of space-time or whether this curvature is itself responsible for the existence of matter." (Gerald J Whitrow, "The Structure of the Universe: An Introduction to Cosmology", 1949)

"The field equation may [...] be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium - in complete analogy with […] the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish." (Howard P Robertson, "Geometry as a Branch of Physics", 1949)

"Formerly, the beginner was taught to crawl through the underbrush, never lifting his eyes to the trees; today he is often made to focus on the curvature of the universe, missing even the earth." (Clifford Truesdell, "Six Lectures on Modern Natural Philosophy", 1966)

"In fact, all our theories of science are formulated on the assumption that space-time is smooth and nearly flat, so they break down at the big bang singularity, where the curvature of space-time is infinite." (Stephen W Hawking, "A Brief History of Time", 1988)

"The fundamental concept of Gauss’s surface theory is the curvature, a quantity that is positive (and constant) for a sphere, zero for the plane and cylinder, and negative for surfaces that are 'saddle-shaped' in the neighborhood of each point." (John Stillwell, The Four Pillars of Geometry, 2000) 

"The simplest surface of constant negative curvature is called the pseudosphere (somewhat misleadingly, because constant curvature is about all it has in common with the sphere). It is more accurately known as the tractroid, because it is the surface of revolution of the curve known as the tractrix. The defining property of the tractrix is that its tangent has constant length a between the curve and the x-axis." (John Stillwell, The Four Pillars of Geometry, 2000) 

"General relativity explains gravitation as a curvature, or bending, or warping, of the geometry of space-time, produced by the presence of matter. Free fall in a space shuttle around Earth, where space is warped, produces weightlessness, and is equivalent from the observer's point of view to freely moving in empty space where there is no large massive body producing curvature. In free fall we move along a 'geodesic' in the curved space-time, which is essentially a straight-line motion over small distances. But it becomes a curved trajectory when viewed at large distances. This is what produces the closed elliptical orbits of planets, with tiny corrections that have been correctly predicted and measured. Planets in orbits are actually in free fall in a curved space-time!" (Leon M Lederman & Christopher T Hill, "Symmetry and the Beautiful Universe", 2004)

"In maps we have scale models of terrain, but projected onto a plane, thus producing occlusion of a sort not inherent to three-dimensional imaging. Maps do not usually have an obvious perspective; but we see perspectivity when, for example, the curvature of the earth makes marginal distortion inevitable as a result of this projection that lowers the dimensionality. A map too is the product of a measuring procedure, but they bring to light a much more important point about ‘point of view’, essentially independent of these limitations in cartography. The point extends to all varieties of modeling, but is made salient by the sense in which use enters the concept of ‘map’ from the beginning. A map is not only an object used to represent features of a terrain, it is an object for the use of the industrial designer, the navigator, and most of all the traveler, to plan and direct action. This brings us to an aspect of scientific representation not touched on so far, though implicit in the discussion of perspective, crucial to its overall understanding: its indexicality."  (Bas C van Fraassen, "Scientific Representation: Paradoxes of Perspective", 2008)

"Curvature is a central concept in differential geometry. There are conceptually different ways to define it, associated with different mathematical objects, the metric tensor, and the affine connection. In our case, however, the affine connection may be derived from the metric. The 'affine curvature' is associated with the notion of parallel transport of vectors as introduced by Levi-Civita. This is most simply illustrated in the case of a two-dimensional surface embedded in three-dimensional space. Let us take a closed curve on that surface and attach to a point on that curve a vector tangent to the surface. Let us now transport that vector along the curve, keeping it parallel to itself. When it comes back to its original position, it will coincide with the original vector if the surface is flat or deviate from it by a certain angle if the surface is curved. If one takes a small curve around a point on the surface, then the ratio of the angle between the original and the final vector and the area enclosed by the curve is the curvature at that point. The curvature at a point on a two-dimensional surface is a pure number." (Hanoch Gutfreund, "The Road to Relativity", 2015)

On Curvature I

"In order to know the curvature of a curve, the determination of the radius of the osculating circle furnishes us the best measure, where for each point of the curve we find a circle whose curvature is precisely the same. However, when one looks for the curvature of a surface, the question is very equivocal and not at all susceptible to an absolute response, as in the case above. There are only spherical surfaces where one would be able to measure the curvature, assuming the curvature of the sphere is the curvature of its great circles, and whose radius could be considered the appropriate measure. But for other surfaces one doesn’t know even how to compare a surface with a sphere, as when one can always compare the curvature of a curve with that of a circle. The reason is evident, since at each point of a surface there are an infinite number of different curvatures. One has to only consider a cylinder, where along the directions parallel to the axis, there is no curvature, whereas in the directions perpendicular to the axis, which are circles, the curvatures are all the same, and all other oblique sections to the axis give a particular curvature. It’s the same for all other surfaces, where it can happen that in one direction the curvature is convex, and in another it is concave, as in those resembling a saddle." (Leonhard Euler, "Recherches sur la courbure des surfaces", 1767)

"Arc, amplitude, and curvature sustain a similar relation to each other as time, motion, and velocity, or as volume, mass, and density." (Carl Friedrich Gauss, 1827)

"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)

"I hold: 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface that is on the average fiat; namely, that the ordinary laws of geometry are not valid in them; 2) that this property of being curved or distorted is constantly being passed on from one portion of space to another after the manner of a wave; 3) that this variation of the curvature of space is what really happens in the phenomenon that we call the motion of matter, whether ponderable or ethereal; 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870) 

"Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870) 

"We may conceive our space to have everywhere a nearly uniform curvature, but that slight variations of the curvature may occur from point to point, and themselves vary with the time. These variations of the curvature with the time may produce effects which we not unnaturally attribute to physical causes independent of the geometry of our space. We might even go so far as to assign to this variation of the curvature of space 'what really happens in that phenomenon which we term the motion of matter'." (William K Clifford, Richard C Rowe & Karl Pearson,"The Common Sense of the Exact Sciences", 1885)

"Mathematics accomplishes really nothing outside of the realm of magnitude; marvellous, however, is the skill with which it masters magnitude wherever it finds it. We recall at once the network of lines which it has spun about heavens and earth; the system of lines to which azimuth and altitude, declination and right ascension, longitude and latitude are referred; those abscissas and ordinates, tangents and normals, circles of curvature and evolutes; those trigonometric and logarithmic functions which have been prepared in advance and await application. A look at this apparatus is sufficient to show that mathematicians are not magicians, but that everything is accomplished by natural means; one is rather impressed by the multitude of skillful machines, numerous witnesses of a manifold and intensely active industry, admirably fitted for the acquisition of true and lasting treasures."(Johann F Herbart, 1890)

"[...]  the illustration of a space of constant positive measure of curvature by the familiar example of the sphere is somewhat misleading.  Owing to the fact that on the sphere the geodesic lines (great circles) issuing from any point all meet again in another definite point, antipodal, so to speak, to the original point, the existence of such an antipodal point has sometimes been regarded as a necessary consequence of the assumption of a constant positive curvature. The projective theory of non-Euclidean space shows immediately that the existence of an antipodal point, though compatible with the nature of an elliptic space, is not necessary, but that two geodesic lines in such a space may intersect in one point if at all." (Felix Klein, "The Most Recent Researches in Non-Euclidian Geometry", [lecture] 1893)

On Continuity II (Topology)

"Things are called continuous when the touching limits of each become one and the same and are contained in each other. Continuity is impossible if these extremities are two. […] Continuity belongs to things that naturally in virtue of their mutual contact form a unity. And in whatever way that which holds them together is one, so too will the whole be one."(Aristotle, "Physics", cca. 350 BC)

"When what surrounds, then, is not separate from the thing, but is in continuity with it, the thing is said to be in what surrounds it, not in the sense of in place, but as a part in a whole. But when the thing is separate or in contact, it is immediately ‘in’ the inner surface of the surrounding body, and this surface is neither a part of what is in it nor yet greater than its extension, but equal to it; for the extremities of things which touch are coincident." (Aristotle, "Physics", cca. 350 BC)

"I hold: 1) that small portions of space are, in fact, of a nature analogous to little hills on a surface that is on the average fiat; namely, that the ordinary laws of geometry are not valid in them; 2) that this property of being curved or distorted is constantly being passed on from one portion of space to another after the manner of a wave; 3) that this variation of the curvature of space is what really happens in the phenomenon that we call the motion of matter, whether ponderable or ethereal; 4) that in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." (William K Clifford, "On the Space Theory of Matter", [paper delivered before the Cambridge Philosophical Society, 1870)

"This is just the idea given above: to divide space, cuts that are called surfaces are necessary; to divide surfaces, cuts that are called lines are necessary; to divide lines, cuts that are called points are necessary; we can go no further and a point can not be divided, a point not being a continuum. Then lines, which can be divided by cuts which are not continua, will be continua of one dimension; surfaces, which can be divided by continuous cuts of one dimension, will be continua of two dimensions; and finally space, which can be divided by continuous cuts of two dimensions, will be a continuum of three dimensions." (Henri Poincaré, 1912)

"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)

"In topology we are concerned with geometrical facts that do not even involve the concepts of a straight line or plane but only the continuous connectiveness between points of a figure." (David Hilbert, "Geometry and Imagination", 1952)

"General or point set topology can be thought of as the abstract study of the ideas of nearness and continuity. This is done in the first place by picking out in elementary geometry those properties of nearness that seem to be fundamental and taking them as axioms." (Andrew H Wallace, "Differential Topology: First Steps", 1968)

"The major strength of catastrophe theory is to provide a qualitative topology of the general structure of discontinuities. Its major weakness is that it frequently is not associated with specific models allowing precise quantitative prediction, although such are possible in principle." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)

"[...] 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)

"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)

"A continuous function preserves closeness of points. A discontinuous function maps arbitrarily close points to points that are not close. The precise definition of continuity involves the relation of distance between pairs of points. […] continuity, a property of functions that allows stretching, shrinking, and folding, but preserves the closeness relation among points." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"Topology is the study of geometric objects as they are transformed by continuous deformations. To a topologist the general shape of the objects is of more importance than distance, size, or angle." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"[…] topology is the study of those properties of geometric objects which remain unchanged under bi-uniform and bi-continuous transformations. Such transformations can be thought of as bending, stretching, twisting or compressing or any combination of these." (Lokenath Debnath, "The Legacy of Leonhard Euler - A Tricentennial Tribute", 2010)

"Topology is a geometry in which all lengths, angles, and areas can be distorted at will. Thus a triangle can be continuously transformed into a rectangle, the rectangle into a square, the square into a circle, and so on. Similarly, a cube can be transformed into a cylinder, the cylinder into a cone, the cone into a sphere. Because of these continuous transformations, topology is known popularly as 'rubber sheet geometry'. All figures that can be transformed into each other by continuous bending, stretching, and twisting are called 'topologically equivalent'." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Topology is an elastic version of geometry that retains the idea of continuity but relaxes rigid metric notions of distance." (Samuel Eilenberg)