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

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish  (1975)  pp. 1–5</TD></TR></table>