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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