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:
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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) |
Comments
References
[a1] | M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1975) pp. 1–5 |
Spherical map. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spherical_map&oldid=18244