Search results
Results from the WOW.Com Content Network
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.
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]
In the theory of smooth manifolds, a congruence is the set of integral curves defined by a nonvanishing vector field defined on the manifold. Congruences are an important concept in general relativity, and are also important in parts of Riemannian geometry.
However, if the action is free and proper, then / has a unique smooth structure such that the projection / is a submersion (in fact, / is a principal -bundle). [ 2 ] The fact that M / G {\displaystyle M/G} is Hausdorff depends only on the properness of the action (as discussed above); the rest of the claim requires freeness and is a consequence ...
In mathematics, differential topology is the field dealing with the topological properties and smooth properties [a] of smooth manifolds.In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape.
Calculus on Manifolds is a brief monograph on the theory of vector-valued functions of several real variables (f : R n →R m) and differentiable manifolds in Euclidean space. . In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats ...
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 ...
Important to applications in mathematics and physics [1] is the notion of a flow on a manifold. In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to .