|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
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.
@ 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.
@ 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].
@ 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.
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