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].