13 Facts About Smooth manifold

In mathematics, a differentiable Smooth manifold is a type of Smooth manifold that is locally similar enough to a vector space to allow one to apply calculus.

In formal terms, a differentiable Smooth manifold is a topological Smooth manifold with a globally defined differential structure.

Any topological Smooth manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space.

Differentiable Smooth manifold is a Hausdorff and second countable topological space, together with a maximal differentiable atlas on.

The tangent bundle of a Smooth manifold is the collection of curves in the Smooth manifold modulo the equivalence relation of first-order contact.

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Every one-dimensional connected smooth manifold is diffeomorphic to either or each with their standard smooth structures.

Pseudo-Riemannian Smooth manifold is a generalization of the notion of Riemannian Smooth manifold where the inner products are allowed to have an indefinite signature, as opposed to being positive-definite; they are still required to be non-degenerate.

Every smooth pseudo-Riemannian and Riemmannian manifold defines a number of associated tensor fields, such as the Riemann curvature tensor.

Not every smooth manifold can be given a pseudo-Riemannian structure; there are topological restrictions on doing so.

Finsler Smooth manifold is a different generalization of a Riemannian Smooth manifold, in which the inner product is replaced with a vector norm; as such, this allows the definition of length, but not angle.

Symplectic Smooth manifold is a Smooth manifold equipped with a closed, nondegenerate 2-form.

Differentiable manifold is then an atlas compatible with the pseudogroup of C functions on R A complex manifold is an atlas compatible with the biholomorphic functions on open sets in C And so forth.

Differentiable Smooth manifold consists of a pair where M is a second countable Hausdorff space, and OM is a sheaf of local R-algebras defined on M, such that the locally ringed space is locally isomorphic to .