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
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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 |
Comments
References
[a1] | J. Nash, "The embedding problem for Riemannian manifolds" Ann. of Math. , 63 (1956) pp. 20–63 |
Immersion of a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Immersion_of_a_manifold&oldid=12393