Immersion of a manifold

A continuous mapping of an -dimensional manifold into an -dimensional manifold such that for each point there exists a neighbourhood in which is an imbedding, i.e. a homeomorphism onto . In particular, if is a homeomorphism into , then it is called an imbedding of in . The immersion is called a -immersion if and are -(smooth) manifolds (, , ) and if the mapping on the corresponding charts is given by functions

that belong to the smoothness class , while the rank of the matrix is equal to at each point (a -(smooth) manifold is a manifold provided with a -structure, where the pseudo-group consists of mappings that are -times differentiable and whose derivatives satisfy the Hölder condition of index ).

The concepts of a surface and a -(smooth) surface are closely related to the concepts of an immersion and a -(smooth) immersion. Two immersions and between manifolds and are called equivalent if there is a homeomorphism such that .

An immersed manifold is a pair consisting of a manifold and an immersion of it. A surface of dimension in a manifold of dimension is a class of equivalent immersions ; each immersion of this class is called a parametrization of the surface. A surface is called -smooth if one can introduce -structures in the manifolds and and if among the parametrizations of the surface one can find a parametrization which in these structures is a -immersion.

The theory of immersed manifolds usually deals with properties that are invariant under the above concept of equivalence, and in essence coincides with the theory of surfaces, particularly when one considers topics related to the geometry of immersions.

Let be a -manifold, , . Any allows for an imbedding into the Euclidean space and a -immersion into for . If is positive and not a power of , then any allows a -imbedding into , whereas for any with there exist closed smooth -dimensional manifolds not allowing even a topological imbedding into (such as, for example, a projective space). If does not have compact components, it allows a -imbedding into .

An orientable -dimensional manifold for allows a -imbedding into . The possibility of immersing an -dimensional manifold into for is related to the Whitney and Pontryagin classes (cf. Pontryagin class) of this manifold. Also, each -smooth -dimensional manifold with , allows a proper immersion into and a proper imbedding into (i.e. an immersion or imbedding such that the pre-image of each compact set is compact). If a Riemannian metric is given on , one frequently considers an isometric immersion of into or into another Riemannian space . A -smooth Riemannian manifold (, ; , ) allows a -smooth isometric immersion into some . In the case of a compact , . Conversely, a -smooth immersion of into (, ) induces a -smooth Riemannian metric on [4].

References

[1]	S. Smale, "The classification of spheres in Euclidean spaces" Ann. of Math. , 69 (1959) pp. 327–344
[2]	H. Jacobowitz, "Implicit function theorems and isometric embeddings" Ann. of Math. , 95 (1972) pp. 191–225
[3]	V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)
[4]	I.Kh. Sabitov, S.Z. Shefel', "The connections between the order of smoothness of a surface and its metric" Sib. Math. J. , 17 : 4 (1976) pp. 687–694 Sibirsk. Mat. Zh. , 17 : 4 (1976) pp. 916–925

References

[a1]	J. Nash, "The embedding problem for Riemannian manifolds" Ann. of Math. , 63 (1956) pp. 20–63