Embeddings and Immersions

Immersion (or local embedding)
$ Def: A map f : S → M 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 R^2n–1.
@ References: Whitney AM(44), AM(44); Hirsch TAMS(59); Smale AM(59); Spring BAMS(05) [history, 1959–1973].

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 n (n + 1) dimensions.
@ References: Pakzad JDG(04) [Sobolev space of immersions]; Ranjbar-Motlagh JGP(09) [non-existence theorem].

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 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 a 6D Minkowski].

Embedding > s.a. foliations; Hypersurface; Whitney Duality Theorem; [graphs; knots].
$ Def: A map f : S → M 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.
@ With metric: Carter CM(97)ht-in, ht/97-in [formalism]; Pavsic & 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 ap/07-in; > s.a. reissner-nordström spacetime.

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 C^p Riemannian manifold with p > 2 has a C^p isometric embedding in n (3n + 11) dimensional Euclidean space; Any non-compact one in n (n + 1) (3n + 11) dimensions (often much less).
@ References: Greene 70.

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 = n (3n + 11) in the compact case (46 for n = 4), and q = (1/6) 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 a0812 [globally hyperbolic]; Kim CQG(09) [with a non-compact Cauchy surface].
@ Hyperspace: Kuchar 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].
@ 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^infty 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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