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−l: f(W) → W is also a diffeomorphism. Further dim E = dim F, and Dfx has rank dim E, for each x ∈ E. We split this definition into two parts.

First, suppose that W is an open subset of a Euclidean space Ed of dimension d, and that j is a continuously differentiable injective mapping of W onto a subset j(W) of a Euclidean space Fd+n of dimension d + n, where n ≥ 0.