## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

19

Differential manifolds in Euclidean space

19.1 Differential manifolds in Euclidean space

A manifold is a topological space which is locally like Euclidean space: each point has an open neighbourhood which is homeomorphic to an open subset of a Euclidean space. A differential manifold is one for which the homeomorphisms can be taken to be diffeomorphisms. We consider differential manifolds which are subspaces of Euclidean space.

Recall that a diffeomorphism f of an open subset W of a Euclidean space E onto a subset f(W) of a Euclidean space F is a bijection of W onto f(W) which is continuously differentiable, and has the property that the derivative Dfx is invertible, for each xW. If so, then f(W) is open in F, and the mapping f1 : f

## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

No credit card required