Immersion of a manifold

A continuous mapping $ F: M ^ {m} \rightarrow N ^ {n} $ of an $ m $- dimensional manifold $ M ^ {m} $ into an $ n $- dimensional manifold $ N ^ {n} $ such that for each point $ x \in M ^ {m} $ there exists a neighbourhood $ U _ {x} $ in which $ F $ is an imbedding, i.e. a homeomorphism onto $ F ( U _ {x} ) \subset N ^ {n} $. In particular, if $ F $ is a homeomorphism into $ F ( M ^ {m} ) $, then it is called an imbedding of $ M ^ {m} $ in $ N ^ {n} $. The immersion $ F $ is called a $ C ^ {l, \alpha } $- immersion if $ M ^ {m} $ and $ N ^ {n} $ are $ C ^ {l, \alpha } $-( smooth) manifolds ( $ l \geq 1 $, $ 0 \leq \alpha < 1 $, $ m \leq n $) and if the mapping $ F $ on the corresponding charts is given by functions

$$ x ^ {i} = f ^ { i } ( u ^ {1} \dots u ^ {m} ), \ i = 1 \dots n, $$

that belong to the smoothness class $ C ^ {l, \alpha } $, while the rank of the matrix $ \| df ^ { i } / du ^ {j} \| $ is equal to $ m $ at each point $ x \in M ^ {m} $( a $ C ^ {l, \alpha } $-( smooth) manifold is a manifold provided with a $ \Gamma $- structure, where the pseudo-group consists of mappings that are $ l $- times differentiable and whose derivatives satisfy the Hölder condition of index $ \alpha $).

The concepts of a surface and a $ C ^ {l, \alpha } $-( smooth) surface are closely related to the concepts of an immersion and a $ C ^ {l, \alpha } $-( smooth) immersion. Two immersions $ F $ and $ G $ between manifolds $ M $ and $ N $ are called equivalent if there is a homeomorphism $ \Phi : M \rightarrow M $ such that $ F = G \Phi $.

An immersed manifold is a pair consisting of a manifold $ M $ and an immersion $ F $ of it. A surface of dimension $ m $ in a manifold $ N ^ {n} $ of dimension $ n $ is a class of equivalent immersions $ F: M ^ {m} \rightarrow N ^ {n} $; each immersion of this class is called a parametrization of the surface. A surface is called $ C ^ {l, \alpha } $- smooth if one can introduce $ C ^ {l, \alpha } $- structures in the manifolds $ M $ and $ N $ and if among the parametrizations of the surface one can find a parametrization $ F $ which in these structures is a $ C ^ {l, \alpha } $- 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 $ M ^ {m} $ be a $ C ^ {l, \alpha } $- manifold, $ l \geq 1 $, $ 0 \leq \alpha < 1 $. Any $ M ^ {m} $ allows for $ m \geq 1 $ an imbedding into the Euclidean space $ \mathbf R ^ {2m} $ and a $ C ^ {l, \alpha } $- immersion into $ \mathbf R ^ {2m-} 1 $ for $ m \geq 2 $. If $ m $ is positive and not a power of $ 2 $, then any $ M ^ {m} $ allows a $ C ^ {l, \alpha } $- imbedding into $ \mathbf R ^ {2m-} 1 $, whereas for any $ m = 2 ^ {s} $ with $ s \geq 0 $ there exist closed smooth $ m $- dimensional manifolds not allowing even a topological imbedding into $ \mathbf R ^ {2m-} 1 $( such as, for example, a projective space). If $ M ^ {m} $ does not have compact components, it allows a $ C ^ {l, \alpha } $- imbedding into $ \mathbf R ^ {2m-} 1 $.

An orientable $ m $- dimensional manifold for $ m \neq 1, 4 $ allows a $ C ^ {l, \alpha } $- imbedding into $ \mathbf R ^ {2m-} 1 $. The possibility of immersing an $ m $- dimensional manifold into $ \mathbf R ^ {n} $ for $ n < 2m- 1 $ is related to the Whitney and Pontryagin classes (cf. Pontryagin class) of this manifold. Also, each $ C ^ {l, \alpha } $- smooth $ m $- dimensional manifold with $ l \geq 1 $, $ 0 \leq \alpha < 1 $ allows a proper immersion into $ \mathbf R ^ {2m} $ and a proper imbedding into $ \mathbf R ^ {2m+} 1 $( i.e. an immersion or imbedding such that the pre-image of each compact set is compact). If a Riemannian metric is given on $ M ^ {m} $, one frequently considers an isometric immersion of $ M ^ {m} $ into $ \mathbf R ^ {n} $ or into another Riemannian space $ N ^ {n} $. A $ C ^ {l, \alpha } $- smooth Riemannian manifold ( $ l = 2 $, $ 0 < \alpha < 1 $; $ l > 2 $, $ 0 \leq \alpha < 1 $) allows a $ C ^ {l, \alpha } $- smooth isometric immersion into some $ \mathbf R ^ {n} $. In the case of a compact $ M ^ {m} $, $ n = ( 2m+ 1)( 6m+ 14) $. Conversely, a $ C ^ {l, \alpha } $- smooth immersion of $ M ^ {m} $ into $ \mathbf R ^ {n} $( $ l \geq 2 $, $ 0 < \alpha < 1 $) induces a $ C ^ {l, \alpha } $- smooth Riemannian metric on $ M ^ {m} $[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