Immersions of Manifolds |

**Immersion** (or local embedding) > s.a. embeddings.

$ __Def__: A map *f* : *S* → *M* between two differentiable
manifolds is an immersion if (*f*_{*})_{p} is injective
for all *p* ∈ *S*.

* __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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