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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.
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.
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The flow in manifolds is extensively encountered in many industrial processes when it is necessary to distribute a large fluid stream into several parallel streams and then to collect them into one discharge stream, such as fuel cells, plate heat exchanger, radial flow reactor, and irrigation. Manifolds can usually be categorized into one of ...
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