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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).
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)} .
In mathematics, stochastic analysis on manifolds or stochastic differential geometry is the study of stochastic analysis over smooth manifolds. It is therefore a synthesis of stochastic analysis (the extension of calculus to stochastic processes ) and of differential geometry .
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 {(), …, ()} =.
In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to . More precisely, given p ∈ M {\displaystyle p\in M} one is interested in curves γ p : R → M {\displaystyle \gamma _{p}:\mathbb {R} \to M} such that:
A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold is an atlas for such that each transition function is a smooth map, and two smooth atlases for are smoothly equivalent provided their union is again a smooth atlas for .
A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into are isotopic (see Knot theory#Higher dimensions). This is proved using general position, which also allows to show that any two embeddings of an n-manifold into + are isotopic. This result is an isotopy version of the weak Whitney embedding theorem.
Choose a Riemannian metric on the manifold N and let H be the associated kinetic energy. Then the level set H = 1/2 is the unit cotangent bundle of N, a smooth manifold of dimension 2n − 1 fibering over N with fibers being spheres. Then the Liouville form restricted to the unit cotangent bundle is a contact structure.