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].
> Online resources: see Wikipedia page.

Immersion with Riemannian Metric > s.a. 2D manifolds.
* 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.
@ References: Pakzad JDG(04) [Sobolev space of immersions]; Ranjbar-Motlagh JGP(09) [non-existence theorem]; Lawn & Roth DG&A(10) [isometric immersions of hypersurfaces in 4D manifolds using spinors].

Immersion with Indefinite Signature Metric
* 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.
@ General references: Friedman RMP(65); Graves TAMS(79); Xia JGP(09) [Ribaucour transformation and permutability theorem].
@ And general relativity: Estabrook & Wahlquist CQG(93); Edelen CQG(02) [Einstein-Riemann spaces in 6D flat]; Monte IJMPA(09)-a0908 [re Schwarzschild immersion into 6D Minkowski space].

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