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

(see [1]–[3]).

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)

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 -dimensional surfaces in -dimensional Euclidean space" Sib. Math. J. , 10 (1969) pp. 328–333 Sibirsk. Mat. Zh. , 10 : 2 (1969) pp. 459–466
[5]	V.V. Glazyrin, "Topological and metric properties of -saddle surfaces" Soviet Math. Dokl. , 18 (1977) pp. 532–534 Dokl. Akad. Nauk SSSR , 233 : 6 (1977) pp. 1028–1030
[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 -dimensional surfaces of nonpositive extrinsic curvature in a Riemannian space" Math. USSR Sb. , 33 (1977) pp. 485–499 Mat. Sb. , 104 : 4 (1977) pp. 559–576
[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 " Bull. Amer. Math. Soc. , 83 : 1 (1977) pp. 1–35

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