Spherical map

Gauss map, normal spherical map

A mapping from a smooth orientable (hyper)surface in a space to the (unit) sphere with centre at the origin of . It assigns to a point the point with position vector — the (unit) normal to at . In other words, the spherical map is defined by a multivector constructed from independent vectors tangent to :

(here are local coordinates of the point , , and is the position vector of ). For example, when ,

where is the vector product; this simplest case was examined by C.F. Gauss in 1814. The image under the spherical map is called the spherical image of .

The form

is the inverse image of the metric form of , and is called the third fundamental form of the (hyper)surface . Its corresponding tensor is related to the tensors and of the first and second fundamental forms, respectively, by the relation

while the metric connections corresponding to and are adjoint connections.

As well as the spherical map, it is useful in the case of a (hyper)surface that is uniquely projected onto a certain (hyper)plane to consider the so-called normal map . For a (hyper)surface defined by the equation

(here are Cartesian coordinates in ), is defined thus:

where , so .

For non-orientable (hyper)surfaces, the so-called non-orientable spherical map is used — a mapping from into the elliptic space (which can be interpreted as the set of straight lines that pass through the centre of , i.e. -dimensional projective space): The line perpendicular to the tangent plane to at a point is associated with .

The spherical map characterizes the curvature of a (hyper)surface in a space. Indeed, the ratio of the area elements of the spherical image and the surface itself at the point is equal to the total (or Kronecker or outer) curvature — the product of the principal curvatures of at :

In precisely the same way, the (integral) curvature of a set is equal to the area of its spherical image (i.e. the set ):

(1)

Generalizations of the spherical map.

1) The tangent representation — the spherical map of a submanifold to — is a mapping

where is a Grassmann manifold, defined (here) in the following way. Let be the tangent space to at a point , which can be considered as a (hyper)plane in , while is the -dimensional subspace that passes through the origin of parallel to . The mapping is also called the spherical map. A generalization of formula (1) holds for even:

here , where is the curvature form on , is the analogous form on , and is the image of under the spherical map. The normal map has a dual definition: The point is associated with the orthogonal complement to .

2) A Gauss map of a vector bundle into a vector space , , is an (arbitrary) mapping

from the fibre space that induces a linear monomorphism on each fibre. For the canonical vector bundle (which is the subbundle of the product , of which the total space consists of all possible pairs with ), the mapping is called the canonical Gauss map. For any fibre bundle , every Gauss map is a composition of a canonical Gauss map and a morphism of fibre bundles; a Gauss map exists if and only if a mapping (where is the base of the fibre bundle) exists such that and are isomorphic (in particular, for every vector bundle over a paracompact space there is a Gauss map into ). For submanifolds of a Riemannian space, there are several generalizations of spherical maps.

3) An Efimov map relates to surfaces in a Riemannian space and is an extension of the above-mentioned concept of adjoint connections. It is defined more formally because of the lack of absolute parallelism in and the examination of the analogue of the third fundamental form — the square of the covariant differential of the normal — . The relation between the Gaussian curvatures and proves to be more complex (a consequence of the inhomogeneity, generally speaking, of the Codazzi equations). This relation remains as before, i.e. ; here , are the Gaussian curvatures of the metrics and (in the case of , ), and the previous formula is obtained, where is the exterior curvature of in , for example in the following situation: The normal to is an eigenvector of the Ricci tensor of the space (considered at the points of ), in other words, is one of the principal surfaces of this tensor. This is always the case if is a space of constant curvature.

Finally, the concept of a spherical map is introduced for certain classes of irregular surfaces.

4) The polar mapping is a spherical map from a convex (hyper)surface into that associates to a point the set of all unit vectors, drawn from the origin, that are parallel to the normals of the supporting (hyper)planes to at . Aleksandrov's theorem: The spherical image of every Borel set is measurable, and the integral curvature is a totally-additive function.

References

[1]	V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1948) (In Russian)
[2]	I.Ya. Bakel'man, A.L. Verner, B.E. Kantor, "Introduction to differential geometry "in the large" " , Moscow (1973) (In Russian)
[3]	A.S. Mishchenko, A.T. Fomenko, "A course of differential geometry and topology" , MIR (1988) (Translated from Russian)
[4]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)
[5]	J.T. Schwartz, "Differential geometry and topology" , Gordon & Breach (1968)
[6]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[7]	R.L. Bishop, R.J. Crittenden, "Geometry of manifolds" , Acad. Press (1964)
[8]	L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)
[9]	H. Busemann, "Convex surfaces" , Interscience (1958)

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1975) pp. 1–5