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''Gauss map, normal spherical map''
 
''Gauss map, normal spherical map''
  
A mapping from a smooth orientable (hyper)surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867201.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867202.png" /> to the (unit) sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867203.png" /> with centre at the origin of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867204.png" />. It assigns to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867205.png" /> the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867206.png" /> with position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867207.png" /> — the (unit) normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867208.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s0867209.png" />. In other words, the spherical map is defined by a multivector constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672010.png" /> independent vectors tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672011.png" />:
+
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} | }
 +
 
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672012.png" /></td> </tr></table>
+
(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 $,
  
(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672013.png" /> are local coordinates of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672016.png" /> is the position vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672017.png" />). For example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672018.png" />,
+
$$
 +
\overline{n}\;  =
 +
\frac{[ \overline{x}\; _ {u} , \overline{x}\; _ {v} ] }{| [ \overline{x}\; _ {u} , \overline{x}\; _ {v} ] | }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672019.png" /></td> </tr></table>
+
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672020.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672021.png" />.
+
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
 
The form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672022.png" /></td> </tr></table>
+
$$
 +
d \overline{n}\; {}  ^ {2}  = \gamma _ {ij}  du  ^ {i}  du  ^ {j}
 +
$$
  
is the inverse image of the metric form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672023.png" />, and is called the third fundamental form of the (hyper)surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672024.png" />. Its corresponding tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672025.png" /> is related to the tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672027.png" /> of the first and second fundamental forms, respectively, by the relation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672028.png" /></td> </tr></table>
+
$$
 +
\gamma _ {ij}  = g  ^ {ki} b _ {ik} b _ {jl} ,
 +
$$
  
while the metric connections corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672030.png" /> are [[Adjoint connections|adjoint connections]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672031.png" />. For a (hyper)surface defined by the equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672032.png" /></td> </tr></table>
+
$$
 +
x  ^ {k+} 1  = f( x  ^ {1} \dots x  ^ {k} )
 +
$$
  
(here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672033.png" /> are Cartesian coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672034.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672035.png" /> is defined thus:
+
(here $  x  ^ {i} $
 +
are Cartesian coordinates in $  E  ^ {k+} 1 $),  
 +
$  \widetilde{n}  $
 +
is defined thus:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672036.png" /></td> </tr></table>
+
$$
 +
\widetilde{n}  = \{ p _ {1} \dots p _ {k} \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672037.png" />, so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672038.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672039.png" /> into the elliptic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672040.png" /> (which can be interpreted as the set of straight lines that pass through the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672041.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672042.png" />-dimensional projective space): The line perpendicular to the tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672043.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672044.png" /> is associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672045.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672046.png" /> and the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672047.png" /> itself at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672048.png" /> is equal to the total (or Kronecker or outer) curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672049.png" /> — the product of the principal curvatures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672050.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672051.png" />:
+
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 $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672052.png" /></td> </tr></table>
+
$$
 +
K _ {l}  =
 +
\frac{dS  ^  \star  }{dS}
 +
,\ \
 +
\textrm{ i.e. }  K( ds  ^  \star  )  =  K(
 +
\frac{ds)}{K _ {l} }
 +
.
 +
$$
  
In precisely the same way, the (integral) curvature of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672053.png" /> is equal to the area of its spherical image (i.e. the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672054.png" />):
+
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} $):
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672055.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\int\limits \int\limits K _ {l}  dS  = \int\limits \int\limits  dS  ^  \star  .
 +
$$
  
 
==Generalizations of the spherical map.==
 
==Generalizations of the spherical map.==
  
 +
1) The tangent representation — the spherical map of a submanifold  $  M  ^ {k} $
 +
to  $  E  ^ {N} $—
 +
is a mapping
  
1) The tangent representation — the spherical map of a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672056.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672057.png" /> — is a mapping
+
$$
 
+
M  ^ {k}  \rightarrow  G _ {k,N }  ,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672058.png" /></td> </tr></table>
+
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672059.png" /> is a [[Grassmann manifold|Grassmann manifold]], defined (here) in the following way. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672060.png" /> be the tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672061.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672062.png" />, which can be considered as a (hyper)plane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672063.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672064.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672065.png" />-dimensional subspace that passes through the origin of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672066.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672067.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672068.png" /> is also called the spherical map. A generalization of formula (1) holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672069.png" /> even:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672070.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {T _ {N} ( M  ^ {k} ) } \widetilde \Omega    = \int\limits _ {M  ^ {k} } \Omega ,
 +
$$
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672071.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672072.png" /> is the [[Curvature form|curvature form]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672074.png" /> is the analogous form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672075.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672076.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672077.png" /> under the spherical map. The normal map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672078.png" /> has a dual definition: The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672079.png" /> is associated with the orthogonal complement to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672080.png" />.
+
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]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672081.png" /> into a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672083.png" />, is an (arbitrary) mapping
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672084.png" /></td> </tr></table>
+
$$
 +
g: E( \xi  ^ {k} )  \rightarrow  F  ^ {N}
 +
$$
  
from the fibre space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672085.png" /> that induces a linear monomorphism on each fibre. For the canonical vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672086.png" /> (which is the subbundle of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672087.png" />, of which the total space consists of all possible pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672088.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672089.png" />), the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672090.png" /> is called the canonical Gauss map. For any fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672091.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672092.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672093.png" /> is the base of the fibre bundle) exists such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672095.png" /> are isomorphic (in particular, for every vector bundle over a paracompact space there is a Gauss map into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672096.png" />). For submanifolds of a Riemannian space, there are several generalizations of spherical maps.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672097.png" /> in a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672098.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s08672099.png" /> and the examination of the analogue of the third fundamental form — the square of the covariant differential of the normal — <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720100.png" />. The relation between the Gaussian curvatures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720102.png" /> proves to be more complex (a consequence of the inhomogeneity, generally speaking, of the Codazzi equations). This relation remains as before, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720103.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720105.png" /> are the Gaussian curvatures of the metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720107.png" /> (in the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720109.png" />), and the previous formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720110.png" /> is obtained, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720111.png" /> is the exterior curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720112.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720113.png" />, for example in the following situation: The normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720114.png" /> is an eigenvector of the [[Ricci tensor|Ricci tensor]] of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720115.png" /> (considered at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720116.png" />), in other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720117.png" /> is one of the principal surfaces of this tensor. This is always the case if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720118.png" /> is a space of constant curvature.
+
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720119.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720120.png" /> that associates to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720121.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720122.png" /> of all unit vectors, drawn from the origin, that are parallel to the normals of the supporting (hyper)planes to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720123.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720124.png" />. Aleksandrov's theorem: The spherical image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720125.png" /> of every Borel set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720126.png" /> is measurable, and the integral curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086720/s086720127.png" /> is a totally-additive function.
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:22, 6 June 2020


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)

Comments

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1975) pp. 1–5
How to Cite This Entry:
Spherical map. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Spherical_map&oldid=48778
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article