Geometry of immersed manifolds
A theory that deals with the extrinsic geometry and the relation between the extrinsic and intrinsic geometry (cf. also Interior geometry) of submanifolds in a Euclidean or Riemannian space. The geometry of immersed manifolds is a generalization of the classical differential geometry of surfaces in the Euclidean space . The intrinsic and extrinsic geometries of an immersed manifold are usually described locally by means of the first and the second fundamental form, respectively. For immersions of an
-dimensional manifold
in a manifold
one has a concept of congruence (see Immersion of a manifold). In the geometry of immersed manifolds one examines properties that are identical for congruent immersions, i.e. properties of a surface
defined by an immersion
. In this connection, an immersion and a surface are not distinguished from the geometrical point of view. An immersion
induces a mapping
of the tangent bundles (cf. Tangent bundle).
The first quadratic (fundamental) form of a submanifold
is defined on
by
![]() |
where ,
and
is a Riemannian metric on
. Here and subsequently, the vectors
are not distinguished in symbols from their image
. The quadratic form
defines the structure of a Riemannian space
on
; the properties of
constitute the subject of the intrinsic geometry of the submanifold
. If
,
,
,
, are local coordinates in
and
, the immersion
is given by the parametric equations
. In local coordinates
![]() |
where and
are the components of the vectors
and
,
![]() |
and are the components of the metric tensor
of the Riemannian space
.
Concepts such as curve length, volume of a region, Levi-Civita connection of the intrinsic metric, curvature transformation
, etc., relate to the intrinsic geometry of
. The computational formulas applying here can be consulted in Riemannian geometry.
The second (fundamental) tensor (form) is defined by
![]() |
where and
are the Levi-Civita connections in
and
, respectively. In fact,
is not dependent on the vector fields
and
but only on their values at the point
and is a bilinear symmetric mapping
![]() |
where is the normal bundle of
in
. For each unit vector
, the equations
![]() |
define the second quadratic form (or second fundamental form) and the shape operator
in the direction of
. In local coordinates the components
of the form
are
![]() |
where are the components of
.
One defines the principal curvature, principal direction in the direction of , and other related concepts for the form
in the usual way (i.e. as for a surface in the Euclidean space
).
Using elementary symmetric functions, one can construct various principal-curvature functions, such as, for example, the mean curvature
![]() |
where is an orthonormal set of normals and
are the principal curvatures of the forms
; the Chern–Lashof curvature
![]() |
where is the volume of the sphere
; and also the length of the second fundamental form
![]() |
The values of the first and second fundamental forms for a submanifold at a point define it near
infinitesimally up to small quantities of the second order. Each
,
, corresponds to an osculating paraboloid (for a submanifold in a Euclidean space, this is the osculating paraboloid for the projection of the submanifold on the
-dimensional plane defined by
and
). If
(i.e. in the case of a hypersurface), the form
is unique up to sign. In that case, the second fundamental tensor and the second fundamental form do not differ, and the theory acquires considerable similarity with the classical theory of surfaces in
.
Basic equations.
The basic equations for an immersed manifold, i.e. the Gauss equations, the Codazzi–Mainardi equations and the Ricci equations, relate the first and second fundamental forms and the curvature tensors of and
. For each vector field section
over
of the vector bundle
restricted to
, let
denote the tangential component and
the normal component. The Gauss formula defining the second fundamental form,
![]() |
gives the normal-tangent decomposition of for
vector fields on
. The Weingarten formula (defining the shape operator),
![]() |
does the same in case is a vector field on
and the section
of
is normal to
. In terms of the shape operators and the second fundamental form one finds that for three vector fields
,
,
on
the tangential component of
is equal to
![]() |
Taking a fourth vector field on
this leads to the Gauss equations
![]() | (1) |
![]() |
![]() |
The normal component of is equal to
![]() | (2) |
![]() |
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Define the connection on the vector bundle
, where
is the normal bundle to
in
, by the formula
![]() |
Then (2) can be rewritten as
![]() | (2prm) |
Equations (2) (or (2prm)) are the Codazzi–Mainardi equations (in intrinsic form) (cf. also Peterson–Codazzi equations).
Finally, consider the normal component of where
is a section of
(by the symmetry properties of
the horizontal component follows from the Codazzi–Mainardi equations). One has
![]() | (3) |
the Ricci equations. Here is the curvature tensor of the connection
on the normal bundle
. The Gauss, Codazzi–Mainardi and Ricci equations are the only general equations available for an isometric immersion. It is reasonable to expect something interesting for
if three of the fields are normal. Indeed,
then has nothing to do with the immersed manifold
at the point
(except with the point
itself).
If the ambient manifold is of constant curvature
, then
and so
is tangent to
. The Gauss, Ricci and Codazzi–Mainardi equations reduce to
![]() | (4) |
![]() |
![]() |
![]() | (5) |
where , and
![]() | (6) |
These equations make sense in a more general setting. Indeed, let be a Riemannian vector bundle over
, i.e. there is a (bundle) Riemannian metric
on
, and let there be a Riemannian connection
, where
denotes the space of smooth sections of
, which is adapted to the metric. This last phrase means that
. A bilinear mapping
such that
is self-adjoint for all
is called a second fundamental tensor in
. The associated second fundamental form in
is then defined by
![]() |
The three equations (4), (5), (6) make perfect sense in this more general setting. One has now the following generalization of the Bonnet theorem [2]: Let be a simply-connected Riemannian manifold equipped with a Riemannian vector bundle of dimension
with a compatible connection
, second fundamental tensor
and associated second fundamental form
. Suppose that equations (4), (5), (6) hold. Then there is an isometric immersion of
into a simply-connected Riemannian manifold of constant curvature
(a space form) and dimension
such that the normal bundle is
.
This immersion is unique in the following sense. Let be two isometric immersions of
into a space form
of curvature
and with normal bundles
and
, with their induced bundle metrics, second fundamental forms and connections. Suppose that there is a bundle mapping
,
![]() |
covering an isometry of
and such that
preserves the bundle metrics, connections and second fundamental forms. Then there is a rigid motion
of
such that
.
Immersion classes.
The geometry of higher-dimensional immersed manifolds arose and developed a long time ago from the theory of the existence of isometric immersions of Riemannian manifolds in , or less often in a space of constant curvature
(see Isometric immersion). Concerning extrinsic geometrical properties and the links between the extrinsic and the intrinsic geometry of surfaces, two-dimensional surfaces in
only have been examined in detail. In that case there exists a classification of the points on the surface, leading for two-dimensional surfaces to the classes of convex surfaces, saddle surfaces and developable surfaces. Among others, these classes are basic objects of research in differential geometry in the large. In the higher-dimensional case, no such classification of points on a surface is known (1983). Only certain classes of higher-dimensional surfaces are known:
-convex,
-saddle,
-developable surfaces.
-convex surfaces.
A surface in
is called
-convex if for each point
there exists a normal
for which
is positive definite, and if for any
-dimensional direction
,
, one can find on
a two-dimensional direction
such that
(or
) for each
for
,
. A two-convex surface
in
is a convex hypersurface in some
[4]. The intrinsic metric of a
-convex surface has the following property: At each point
and for each
-dimensional direction
in the tangent space one can find a two-dimensional direction
in which the Riemannian curvature is strictly positive.
-saddle surfaces.
A surface in
is called
-saddle if for each point
and for each normal
the number of eigen values of
of one fixed sign does not exceed
,
. A two-dimensional
-saddle surface is an ordinary saddle surface in
from which one cannot cut off the saddle point with a hyperplane. The intrinsic metric of a
-saddle surface has the following property: At each point
for each
-dimensional direction
in the tangent space there is a two-dimensional direction
in which the Riemannian curvature is not positive. If a
-saddle surface is complete in
, then its homology
for
[4], [5]. A complete
-dimensional
-saddle surface
with non-negative Ricci curvature is a cylinder with generator of dimension
.
-developable (
-parabolic) surfaces.
A surface in
is called
-developable if for each point
there exists a
-dimensional direction
that consists of eigen vectors belonging to the zero eigen value of the second fundamental form with respect to each normal at the given point. The intrinsic metric of a
-developable surface has the following property: At each point
one can find a
-dimensional subspace
of the tangent space
such that
for any vector
, where
is any vector in the tangent space and
is the curvature operator. If a
-developable surface
is complete in
and carries an intrinsic metric of non-positive Ricci curvature, then it is a cylinder with a
-dimensional generator .
Free immersions.
If the image of has maximum possible dimension
at each point
, then the immersion is called free. In that case, the first and second derivatives of the radius vector of the immersion
form a linearly independent system. In the class of free immersions there exist isometric immersions of dimension
, giving rise to a complete loss of the connection between the intrinsic and the extrinsic geometry. For example, two free isometric immersions of an
-dimensional manifold
in
,
, may be connected by a homotopy consisting of free isometric immersions of
[7].
Immersions with small codimension.
If the codimension of an immersion is small, then it follows from the conditions on the intrinsic metric of the manifold that there must be restrictions on the second fundamental form of the surface. Also, the properties of the second fundamental form enable one to derive topological and extrinsic geometrical properties for the surface. In particular, one obtains non-immersibility theorems. For example, if an
with sectional curvature
is isometrically immersed in
with
, then
is a
-saddle surface and its homology (in the case of completeness)
vanishes for
[5]. In particular, a compact
with
cannot be immersed in
[8], [9]. If on the other hand
, then
is not even locally immersible in
[9]. Similarly, an
with
is not immersible in the sphere
of radius 1. A compact
in
has Euler characteristic zero and a compact parallelizable covering manifold if
[10]. Regarding a surface
in
for
and
, it is known that its normal Pontryagin classes (cf. Pontryagin class) satisfy the conditions
![]() |
If , it follows from
that
is a
-convex surface [9]. In particular, for
it is a two-convex surface. If
and
, a compact surface
with
has the homologies of a sphere [11]. If
in
has non-positive sectional curvature, then it is an
-developable surface and, in the case of completeness,
is a cylinder with generator of dimension
[10]. If, on the other hand,
and
, the immersion of the manifold
in
is a
-developable surface [8], and in the case of completeness
is a cylinder with generator of dimension
. Under more general assumptions, a compact surface
![]() |
is a product of hypersurfaces [12].
References
[1] | L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949) |
[2] | B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973) |
[3] | S.-S. Chern, R.K. Lashof, "On the total curvature of immersed manifolds" Amer. J. Math. , 79 (1957) pp. 306–318 |
[4] | S.Z. Shefel', "Two classes of ![]() ![]() |
[5] | V.V. Glazyrin, "Topological and metric properties of ![]() |
[6a] | P. Hartman, "On isometric immersions in Euclidean space of manifolds with non-negative sectional curvatures" Trans. Amer. Math. Soc. , 115 (1965) pp. 94–109 |
[6b] | P. Hartman, "On the isometric immersions in Euclidean space of manifolds with nonnegative sectional curvatures II" Trans. Amer. Math. Soc. , 147 (1970) pp. 529–540 |
[7] | M.L. Gromov, "Isometric imbeddings and immersions" Soviet Math. Dokl. , 11 (1970) pp. 794–797 Dokl. Akad. Nauk SSSR , 192 : 6 (1970) pp. 1206–1209 |
[8] | S.-S. Chern, N.H. Kuiper, "Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space" Ann. of Math. , 56 : 3 (1952) pp. 422–430 |
[9] | Yu.E. Borovskii, S.Z. Shefel', "On Chern–Kuiper theorem" Sib. Math. J. , 19 (1978) pp. 978 Sibirsk. Mat. Zh. , 19 : 6 (1978) pp. 1386–1387 |
[10] | A.A. Borisenko, "Complete ![]() |
[11] | J.D. Moore, "Codimension two submanifolds of positive curvature" Proc. Amer. Math. Soc. , 70 : 1 (1978) pp. 72–74 |
[12] | R.B. Gardner, "New viewpoints in the geometry of submanifolds of ![]() |
Comments
The mean curvature is the length of the mean curvature normal which is given by the trace of the bilinear (normal vector-valued) form with respect to the first fundamental form. Minimal immersions are defined by the property that the mean curvature normal vanishes identically. They locally solve the least area problem.
The integrand of the Chern–Lashof curvature is the absolute value of the Lipschitz–Killing curvature. The well-known Chern–Lashof theorem states that the Chern–Lashof curvature is at least 2 for compact immersed submanifolds of Euclidean spaces, and the value 2 is obtained just for convex hyperspheres of affine subspaces of suitable dimensions.
References
[a1] | M. Gromov, "Partial differential relations" , Springer (1986) (Translated from Russian) |
[a2] | M. Gromov, V. Rokhlin, "Embeddings and immersions in Riemannian geometry" Russian Math. Surveys , 25 : 5 (1970) pp. 1–57 Uspekhi Mat. Nauk , 25 : 5 (1970) pp. 3–62 |
Geometry of immersed manifolds. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Geometry_of_immersed_manifolds&oldid=47093