Embeddings of Manifolds |

**In General** > s.a. foliations;
Hypersurface; immersions.

$ __Def__: A map *f* : *S* →
*M* between two differentiable manifolds is an embedding if it is an injective immersion.

* __Idea__: The map *f* a globally one-to-one
immersion, and *f*(*S*) does not intersect itself in *M*.

* __In addition__: Sometimes one wants *S*
to be homeomorphic to *f*(*S*) in the induced topology from *M*.

* __Whitney (strong) embedding theorem__:
Any smooth (Hausdorff, second-countable) *n*-dimensional manifold can be smoothly
embedded in 2*n*-dimensional Euclidean space;
> s.a. Wikipedia page.

@ __General references__: Skopenkov T&A(10)
[classification of smooth embeddings of 4-manifolds in \(\mathbb R\)^{7}];
Daverman & Venema 09.

@ __With metric__: Carter CM(97)ht-fs,
ht/97-ln [formalism];
Pavšič & Tapia gq/00 [references];
> s.a. membranes [dynamics].

@ __Embedding diagrams__: Romano & Price CQG(95)gq/94 [initial data for black hole collisions];
Lu & Suen GRG(03) [extrinsic-curvature-based];
Hledík et al AIP(06)ap/07;
> s.a. reissner-nordström spacetime;
schwarzschild geometry.

> __Related topics__: see knots;
types of graphs [embedded in manifolds]; Whitney
Duality Theorem; Wild Embeddings.

> __Online resources__: see
Wikipedia page.

**Embedding with Riemannian Metric** > s.a. riemannian
geometry / extrinsic curvature.

* __Results__: Any compact *n*-dimensional
C^{1} Riemannian
manifold (with or without boundary) has a C^{1} isometric
embedding in 2*n*-dimensional Euclidean space; Any non-compact one in 2*n* + 1
dimensions; However, if a compact one has a C^{1} embedding in
*k* > *n* dimensions, then it also has a C^{1}
isometric embedding there (thus any point has a neighborhood with a C^{1}
isometric embedding in *n* + 1 dimensions).

* __Ideal embeddings__: The embedded manifold
receives the least amount of tension from the surrounding space.

* __Results__: Any compact, *n*-dimensional
C^{p} Riemannian manifold with *p* > 2 has
a C^{p} isometric embedding in \(1\over2\)*n*
(3*n* + 11) dimensional Euclidean space; Any non-compact one in \(1\over2\)*n*
(*n* + 1) (3*n* + 11) dimensions (often much less).

@ __References__: Greene 70;
Arnlind et al a1001 [geometry and algebraic structure];
Arnlind et al a1003 [in terms of Nambu brackets].

**Embedding with Lorentzian Metric** > s.a. lorentzian
geometry [hypersurfaces]; extrinsic curvature;
formulations of general relativity.

* __Remark__: Obviously,
the Lorentzian, global case in general is not so easy; For example, the metric
may have closed timelike curves.

* __In flat spaces__: Any
C^{k}
Lorentzian manifold, with 3 ≤ *k *< ∞,
can be embedded in a (*q* + 2)-dimensional flat space (2 are timelike!), with
*q* = \(1\over2\)*n* (3*n* +
11) in the compact case (46 for *n* =
4), and *q* = \(1\over6\)*n* (2*n*^{2} +
37) + (5/2) *n*^{2} + 1 in the
non-compact case (87 for *n* = 4); If the spacetime is
globally hyperbolic, *q *+ 1 is enough.

* __In Ricci-flat spaces__:
(in 4D, Campbell-Magaard theorem) Any *n*-dimensional (*n* ≥ 3)
Lorentzian manifold can be isometrically and harmonically embedded in a (*n*
+ 1)-dimensional semi-Riemannian Ricci-flat space.

* __Hyperspace__: In general
relativity, the space of embeddings of a hypersurface in spacetime (roughly!).

@ __General references__: Rosen RMP(65)
[examples]; Clarke PRS(70);
Greene 70; Mueller & Sánchez TAMS-a0812 [globally
hyperbolic]; Kim CQG(09)
[with a non-compact Cauchy surface]; Ponce de León CQG(15)-a1509;
Sheykin et al a2004 [explicit embeddings].

@ __Hyperspace__: Kuchař JMP(76),
JMP(76),
JMP(76),
JMP(77).

@ __For 4D Ricci-flat spaces__: Romero et al GRG(96),
Lidsey et al CQG(97)gq/99 [4D solution in 5D];
Mashhoon & Wesson GRG(07) [with a 4D cosmological constant].

@ __For 4D spaces with cosmological constant__: Ponce de León G&C(08)-a0709 [in various 5D spaces].

@ __Campbell-Magaard theorem__: Dahia & Romero JMP(02);
Anderson gq/04 [attack];
Dahia & Romero CQG(05)gq [interpretation];
Wesson gq/05 [apology];
Avalos et al JMP(17)-a1701 [extension to Weyl manifolds].

@ __For n-dimensional Ricci-flat spaces__:
Seahra & Wesson CQG(03)gq;
Chervon et al PLA(04);
Anderson gq/04;
Avalos et al a1708.

@

@

**Embedding with Arbitrary Metric, Global**

* __Results__: For a C^{∞} compact
manifold (with possibly degenerate metric), an embedding can be found in 2*k* = *n* (*n*+5)
dimensions, signature (*k*, *k*), and 2*k* = 2 (2*n*+1) (2*n*+6) dimensions,
signature (*k*, *k*), in the non-compact case.

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send feedback and suggestions to bombelli at olemiss.edu – modified 13 apr 2020