"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)
No comments:
Post a Comment