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Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in R k to M. The group C k (M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain complex.
Introduction to smooth manifolds. New York: Springer. ISBN 0-387-95448-1. A textbook on manifold theory. See also the same author's textbooks on topological manifolds (a lower level of structure) and Riemannian geometry (a higher level of structure).
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations.The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric.
Stochastic differential geometry provides insight into classical analytic problems, and offers new approaches to prove results by means of probability. For example, one can apply Brownian motion to the Dirichlet problem at infinity for Cartan-Hadamard manifolds [4] or give a probabilistic proof of the Atiyah-Singer index theorem. [5]
For smooth manifolds M the problem reduces to finding the form of the homomorphism () (), where () is the oriented bordism group of X. [4] The connection between the bordism groups and the Thom spaces MSO(k) clarified the Steenrod problem by reducing it to the study of the homomorphisms ( ()) ().
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold.It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology.
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Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. [1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential of a ...