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Let X and Y be oriented smooth closed manifolds, and f: X → Y a continuous map. Let v f =f * (TY) − TX in the K-group K(X). If dim(X) ≡ dim(Y) mod 2, then (()) = (() / ^ ()),where ch is the Chern character, d(v f) an element of the integral cohomology group H 2 (Y, Z) satisfying d(v f) ≡ f * w 2 (TY)-w 2 (TX) mod 2, f K* the Gysin homomorphism for K-theory, and f H* the Gysin ...
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.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume.
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 .
The n is the number of ports and L the length of the manifold (Fig. 2). This is fundamental of manifold and network models. Thus, a T-junction (Fig. 3) can be represented by two Bernoulli equations according to two flow outlets. A flow in manifold can be represented by a channel network model.
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Theorem: Every smooth manifold admits a (non-canonical) Riemannian metric. [13] This is a fundamental result. Although much of the basic theory of Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are Hausdorff and paracompact.