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Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity.
If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N. In differential geometry , pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces.
Let be a smooth manifold; a (smooth) distribution assigns to any point a vector subspace in a smooth way. More precisely, consists of a collection {} of vector subspaces with the following property: Around any there exist a neighbourhood and a collection of vector fields, …, such that, for any point , span {(), …, ()} =.
Let :, (,) be a (left) group action of a Lie group on a smooth manifold ; it is called a Lie group action (or smooth action) if the map is differentiable. Equivalently, a Lie group action of G {\displaystyle G} on M {\displaystyle M} consists of a Lie group homomorphism G → D i f f ( M ) {\displaystyle G\to \mathrm {Diff} (M)} .
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 homomorphism for cohomology . [1] This theorem was first proven by Atiyah and Hirzebruch. [2] The theorem is proven by considering several ...
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.
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Important to applications in mathematics and physics [1] is the notion of a flow on a manifold. In particular, if M {\displaystyle M} is a smooth manifold and X {\displaystyle X} is a smooth vector field , one is interested in finding integral curves to X {\displaystyle X} .