Immersions of Manifolds

Immersion (or local embedding) > s.a. embeddings.
\$ Def: A map f : SM between two differentiable manifolds is an immersion if (f*)p is injective for all pS.
* Idea: This means that f is locally 1-to-1, but there may be self-intersections globally, as opposed to what happens with an embedding; One usually requires also that f and its local inverse be smooth.
* Results: Any smooth compact n-manifold can be immersed in $$\mathbb R$$2n–1.
@ References: Whitney AM(44), AM(44); Hirsch TAMS(59); Smale AM(59); Spring BAMS(05) [history, 1959–1973].
* Results: Any n-dimensional Riemannian manifold with analytic metric can be analytically and isometrically locally embedded in a Euclidean space of $$1\over2$$n (n + 1) dimensions.
* Results: Any pseudo-Riemannian manifold of signature (p, q), p + q = n, with analytic metric can be analytically and isometrically embedded in a flat space of dimension $$1\over2$$n (n + 1) and signature (r, s), r > p, s > q.