Embeddings of Manifolds| 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 2n-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
C1 Riemannian
manifold (with or without boundary) has a C1 isometric
embedding in 2n-dimensional Euclidean space; Any non-compact one in 2n + 1
dimensions; However, if a compact one has a C1 embedding in
k > n dimensions, then it also has a C1
isometric embedding there (thus any point has a neighborhood with a C1
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
Cp Riemannian manifold with p > 2 has
a Cp isometric embedding in \(1\over2\)n
(3n + 11) dimensional Euclidean space; Any non-compact one in \(1\over2\)n
(n + 1) (3n + 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
Ck
Lorentzian manifold, with 3 ≤ k < ∞,
can be embedded in a (q + 2)-dimensional flat space (2 are timelike!), with
q = \(1\over2\)n (3n +
11) in the compact case (46 for n =
4), and q = \(1\over6\)n (2n2 +
37) + (5/2) n2 + 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.
@ Codimension-1 embeddings: Anderson & Lidsey CQG(01)gq,
Katzourakis mp/04,
m.DG/05 [in Einstein spaces];
Dahia & Romero JMP(02) [with prescribed D+1 Ricci tensor];
Haesen & Verstraelen JMP(04)gq/03 [ideal embeddings];
Kuhfittig AP(18)-a1805 [applications to wormholes and galaxy rotation curves].
@ Codimension-2 embeddings: Dillen et al JGP(04) [inequalities intrinsic/extrinsic curvature].
Embedding with Arbitrary Metric, Global
* Results: For a C∞ compact
manifold (with possibly degenerate metric), an embedding can be found in 2k = n (n+5)
dimensions, signature (k, k), and 2k = 2 (2n+1) (2n+6) dimensions,
signature (k, k), in the non-compact case.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 13 apr 2020