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