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 x ∈ W. If so, then f(W) is open in F, and the mapping f –1 : f