Spherical map

Gauss map, normal spherical map

A mapping from a smooth orientable (hyper)surface $ M ^ {k} $ in a space $ E ^ {k+} 1 $ to the (unit) sphere $ S ^ {k} $ with centre at the origin of $ E ^ {k+} 1 $. It assigns to a point $ x \in M ^ {k} $ the point $ x ^ \star \in S ^ {k} $ with position vector $ \overline{n}\; ( x) $— the (unit) normal to $ M ^ {k} $ at $ x $. In other words, the spherical map is defined by a multivector constructed from $ k $ independent vectors tangent to $ M ^ {k} $:

$$ \overline{n}\; = \frac{\overline{x}\; _ {1} \wedge \dots \wedge \overline{x}\; _ {k} }{| \overline{x}\; _ {1} \wedge \dots \wedge \overline{x}\; _ {k} | } $$

(here $ u ^ {1} \dots u ^ {k} $ are local coordinates of the point $ x $, $ \overline{x}\; _ {i} = ( \partial \overline{x}\; / \partial u ^ {i} ) $, and $ \overline{x}\; $ is the position vector of $ M ^ {k} $). For example, when $ k = 2 $,

$$ \overline{n}\; = \frac{[ \overline{x}\; _ {u} , \overline{x}\; _ {v} ] }{| [ \overline{x}\; _ {u} , \overline{x}\; _ {v} ] | } , $$

where $ [ \cdot , \cdot ] $ 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 $ M ^ {k} $.

The form

$$ d \overline{n}\; {} ^ {2} = \gamma _ {ij} du ^ {i} du ^ {j} $$

is the inverse image of the metric form of $ S ^ {k} $, and is called the third fundamental form of the (hyper)surface $ M ^ {k} $. Its corresponding tensor $ \gamma _ {ij} $ is related to the tensors $ g _ {ij} $ and $ b _ {ij} $ of the first and second fundamental forms, respectively, by the relation

$$ \gamma _ {ij} = g ^ {ki} b _ {ik} b _ {jl} , $$

while the metric connections corresponding to $ g _ {ij} $ and $ \gamma _ {ij} $ 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 $ \widetilde{n} $. For a (hyper)surface defined by the equation

$$ x ^ {k+} 1 = f( x ^ {1} \dots x ^ {k} ) $$

(here $ x ^ {i} $ are Cartesian coordinates in $ E ^ {k+} 1 $), $ \widetilde{n} $ is defined thus:

$$ \widetilde{n} = \{ p _ {1} \dots p _ {k} \} , $$

where $ p _ {i} = ( \partial f/ \partial x ^ {i} ) $, so $ \widetilde{n} = n \sqrt {1 + \sum p _ {i} ^ {2} } $.

For non-orientable (hyper)surfaces, the so-called non-orientable spherical map is used — a mapping from $ M ^ {k} $ into the elliptic space $ El ^ {k} $( which can be interpreted as the set of straight lines that pass through the centre of $ E ^ {k+} 1 $, i.e. $ k $- dimensional projective space): The line perpendicular to the tangent plane to $ M ^ {k} $ at a point $ x $ is associated with $ x \in M ^ {k} $.

The spherical map characterizes the curvature of a (hyper)surface in a space. Indeed, the ratio of the area elements of the spherical image $ dS ^ \star $ and the surface $ dS $ itself at the point $ x \in M ^ {k} $ is equal to the total (or Kronecker or outer) curvature $ K _ {l} $— the product of the principal curvatures of $ M ^ {k} $ at $ x $:

$$ K _ {l} = \frac{dS ^ \star }{dS} ,\ \ \textrm{ i.e. } K( ds ^ \star ) = K( \frac{ds)}{K _ {l} } . $$

In precisely the same way, the (integral) curvature of a set $ F \subset M ^ {k} $ is equal to the area of its spherical image (i.e. the set $ F ^ { \star } = \overline{n}\; ( F ) \subset S ^ {k} $):

$$ \tag{1 } \int\limits \int\limits K _ {l} dS = \int\limits \int\limits dS ^ \star . $$

Generalizations of the spherical map.

1) The tangent representation — the spherical map of a submanifold $ M ^ {k} $ to $ E ^ {N} $— is a mapping

$$ M ^ {k} \rightarrow G _ {k,N } , $$

where $ G _ {k,N } $ is a Grassmann manifold, defined (here) in the following way. Let $ T _ {x} $ be the tangent space to $ M ^ {k} $ at a point $ x $, which can be considered as a (hyper)plane in $ E ^ {N} $, while $ T( x) $ is the $ k $- dimensional subspace that passes through the origin of $ E ^ {N} $ parallel to $ T _ {x} $. The mapping $ x \rightarrow T( x) $ is also called the spherical map. A generalization of formula (1) holds for $ k $ even:

$$ \int\limits _ {T _ {N} ( M ^ {k} ) } \widetilde \Omega = \int\limits _ {M ^ {k} } \Omega , $$

here $ \Omega = \epsilon ^ {i _ {1} \dots i _ {k} } \Omega _ {i _ {1} i _ {2} } \wedge \dots \wedge \Omega _ {i _ {k-} 1 i _ {k} } $, where $ \Omega _ {ij} $ is the curvature form on $ M ^ {k} $, $ \widetilde \Omega $ is the analogous form on $ G _ {k,N } $, and $ T _ {N} ( M ^ {k} ) $ is the image of $ M ^ {k} $ under the spherical map. The normal map $ M ^ {k} \rightarrow G _ {N- k N } $ has a dual definition: The point $ x \in M ^ {k} $ is associated with the orthogonal complement to $ T( x) $.

2) A Gauss map of a vector bundle $ \xi ^ {k} $ into a vector space $ F ^ { N } $, $ k \leq N \leq \infty $, is an (arbitrary) mapping

$$ g: E( \xi ^ {k} ) \rightarrow F ^ {N} $$

from the fibre space $ E( \xi ^ {k} ) $ that induces a linear monomorphism on each fibre. For the canonical vector bundle $ \gamma _ {k} ^ {N} $( which is the subbundle of the product $ ( G _ {N,k} \times \mathbf R ^ {N} , p, G _ {N,k} ) $, of which the total space consists of all possible pairs $ ( V, x) \in G _ {N,k} \times \mathbf R ^ {N} $ with $ x \in V $), the mapping $ ( V, x) \rightarrow x $ is called the canonical Gauss map. For any fibre bundle $ \xi ^ {k} $, 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 $ f: B( \xi ) \rightarrow G _ {N,k} $( where $ B $ is the base of the fibre bundle) exists such that $ \xi $ and $ f ^ { \star } ( \gamma _ {k} ^ {N} ) $ are isomorphic (in particular, for every vector bundle over a paracompact space there is a Gauss map into $ F ^ { \infty } $). For submanifolds of a Riemannian space, there are several generalizations of spherical maps.

3) An Efimov map relates to surfaces $ M ^ {2} $ in a Riemannian space $ V ^ {3} $ 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 $ V ^ {3} $ and the examination of the analogue of the third fundamental form — the square of the covariant differential of the normal — $ ( D \overline{n}\; ) ^ {2} $. The relation between the Gaussian curvatures $ K( ds ^ \star ) $ and $ K( ds) $ proves to be more complex (a consequence of the inhomogeneity, generally speaking, of the Codazzi equations). This relation remains as before, i.e. $ K( | Dn | ) = ( K( ds))/K _ {l} $; here $ K( ds) $, $ K( | Dn | ) $ are the Gaussian curvatures of the metrics $ ds $ and $ | Dn | $( in the case of $ V ^ {3} = E ^ {3} $, $ K( ds) = K _ {l} $), and the previous formula $ K ( | Dn | ) = K( | dn | ) = 1 $ is obtained, where $ K _ {l} $ is the exterior curvature of $ M ^ {2} $ in $ V ^ {3} $, for example in the following situation: The normal to $ M ^ {2} $ is an eigenvector of the Ricci tensor of the space $ V ^ {3} $( considered at the points of $ M ^ {2} $), in other words, $ M ^ {2} $ is one of the principal surfaces of this tensor. This is always the case if $ V ^ {3} $ 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 $ F ^ { k } $ into $ E ^ {k+} 1 $ that associates to a point $ x \in F ^ { k } $ the set $ \nu ( x) $ of all unit vectors, drawn from the origin, that are parallel to the normals of the supporting (hyper)planes to $ F ^ { k } $ at $ x $. Aleksandrov's theorem: The spherical image $ \nu ( A) $ of every Borel set $ A \subset F ^ { k } $ is measurable, and the integral curvature $ K( A) = \mathop{\rm mes} \nu ( A) $ 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