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Difference between revisions of "Affine differential geometry"

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The branch of geometry dealing with the differential-geometric properties of curves and surfaces that are invariant under transformations of the [[Affine group|affine group]] or its subgroups. The differential geometry of equi-affine space has been most thoroughly studied.
 
The branch of geometry dealing with the differential-geometric properties of curves and surfaces that are invariant under transformations of the [[Affine group|affine group]] or its subgroups. The differential geometry of equi-affine space has been most thoroughly studied.
  
In an equi-affine plane any two vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a0109901.png" /> have an invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a0109902.png" /> — the surface area of the parallelogram constructed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a0109903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a0109904.png" />. With the aid of this concept, the invariant parameter
+
In an equi-affine plane any two vectors $  \mathbf a , \mathbf b $
 +
have an invariant $  ( \mathbf a , \mathbf b ) $—  
 +
the surface area of the parallelogram constructed on $  \mathbf a $
 +
and $  \mathbf b $.  
 +
With the aid of this concept, the invariant parameter
  
<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/a/a010/a010990/a0109905.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ { t } _ {0} ^ { t }
 +
| ( \dot{\mathbf r} , \dot{\mathbf r} dot  ) |  ^ {1/3}  dt,
 +
$$
  
known as the equi-affine arc length, can be constructed for a non-rectilinear curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a0109906.png" />. The differential invariant
+
known as the equi-affine arc length, can be constructed for a non-rectilinear curve $  \mathbf r = \mathbf r (t) $.  
 +
The differential invariant
  
<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/a/a010/a010990/a0109907.png" /></td> </tr></table>
+
$$
 +
= \left (
 +
\frac{d  ^ {2} \mathbf r }{ds  ^ {2} }
 +
,\
  
is called the equi-affine curvature of the plane curve. Constant equi-affine curvature characterizes curves of the second order. A natural equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a0109908.png" /> determines a curve up to an equi-affine transformation. The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a0109909.png" /> is directed along the affine normal to a plane curve; the affine normal at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099011.png" />, is the tangent to the locus of the mid-chords of the curve parallel to the tangent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099012.png" />, and coincides with the diameter of the parabola which has third-order contact with the curve at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099013.png" />.
+
\frac{d  ^ {3} \mathbf r }{d s  ^ {3} }
 +
\right )
 +
$$
  
Passing to the general affine group, two more invariants of the curve are considered: the affine arc length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099014.png" /> and the affine curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099015.png" />. They can be expressed in terms of the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099017.png" /> introduced above:
+
is called the equi-affine curvature of the plane curve. Constant equi-affine curvature characterizes curves of the second order. A natural equation  $  k = f(s) $
 +
determines a curve up to an equi-affine transformation. The vector  $  n = {d  ^ {2} \mathbf r } / {d s  ^ {2} } $
 +
is directed along the affine normal to a plane curve; the affine normal at a point  $  M $,
 +
$  k \neq 0 $,
 +
is the tangent to the locus of the mid-chords of the curve parallel to the tangent at  $  M $,
 +
and coincides with the diameter of the parabola which has third-order contact with the curve at  $  M $.
  
<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/a/a010/a010990/a01099018.png" /></td> </tr></table>
+
Passing to the general affine group, two more invariants of the curve are considered: the affine arc length  $  \sigma $
 +
and the affine curvature  $  \kappa $.
 +
They can be expressed in terms of the invariants  $  s $
 +
and  $  k $
 +
introduced above:
  
(In equi-affine geometry, the magnitudes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099020.png" /> themselves are called the affine arc length and the affine curvature, for the sake of brevity.) The centro-affine arc length, centro-affine curvature, equi-centro-affine arc length and equi-centro-affine curvature of a plane curve are constructed in a similar manner.
+
$$
 +
\sigma  = \int\limits k  ^ {1/2}  ds,\ \
 +
\kappa  =
 +
\frac{1}{k  ^ {3/2} }
 +
\cdot
 +
\frac{dk}{ds}
 +
.
 +
$$
  
In equi-affine space it is possible to assign to any three vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099021.png" /> the invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099022.png" />, which is the volume of the oriented parallelepiped defined by these vectors. The natural parameter (equi-affine arc length) of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099023.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099024.png" />) is defined by the formula
+
(In equi-affine geometry, the magnitudes  $  s $
 +
and  $  k $
 +
themselves are called the affine arc length and the affine curvature, for the sake of brevity.) The centro-affine arc length, centro-affine curvature, equi-centro-affine arc length and equi-centro-affine curvature of a plane curve are constructed in a similar manner.
  
<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/a/a010/a010990/a01099025.png" /></td> </tr></table>
+
In equi-affine space it is possible to assign to any three vectors  $  \mathbf a , \mathbf b , \mathbf c $
 +
the invariant  $  ( \mathbf a , \mathbf b , \mathbf c ) $,
 +
which is the volume of the oriented parallelepiped defined by these vectors. The natural parameter (equi-affine arc length) of a curve  $  \mathbf r = \mathbf r (t) $(
 +
$  \mathbf r \in C  ^ {3} $)
 +
is defined by the formula
  
The differential invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099027.png" />, where the primes denote differentiation with respect to the natural parameter, are called, respectively, the equi-affine curvature and the equi-affine torsion of the spatial curve. The study of the curve is reduced to selecting some moving frame; the frame formed by the vectors
+
$$
 +
= \int\limits _ { t } _ {0} ^ { t }
 +
| ( \dot{\mathbf r} , \dot{\mathbf r} dot , \mathbf r ^ { {. . . }  ) |  ^ {1/6}  dt.
 +
$$
  
<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/a/a010/a010990/a01099028.png" /></td> </tr></table>
+
The differential invariants  $  \kappa = ( \mathbf r  ^  \prime  , \mathbf r  ^ {\prime\prime\prime} , \mathbf r  ^ {\prime\prime\prime\prime} ) $,
 +
$  \tau = - ( \mathbf r  ^ {\prime\prime} , \mathbf r  ^ {\prime\prime\prime} , \mathbf r  ^ {\prime\prime\prime\prime} ) $,
 +
where the primes denote differentiation with respect to the natural parameter, are called, respectively, the equi-affine curvature and the equi-affine torsion of the spatial curve. The study of the curve is reduced to selecting some moving frame; the frame formed by the vectors
 +
 
 +
$$
 +
\left \{ \dot{\mathbf r} , \dot{\mathbf r} dot , \mathbf r ^ { {. . . } + \kappa
 +
\mathbf r
 +
\frac{dot}{4}
 +
\right \}
 +
$$
  
 
and defined by the fourth-order differential neighbourhood of the curve being studied, is especially important. The centro-affine theory of spatial curves has been developed [[#References|[5]]].
 
and defined by the fourth-order differential neighbourhood of the curve being studied, is especially important. The centro-affine theory of spatial curves has been developed [[#References|[5]]].
  
The following tensor is constructed for a non-developable surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099029.png" /> in equi-affine space:
+
The following tensor is constructed for a non-developable surface $  \mathbf r = \mathbf r (u  ^ {1} , u  ^ {2} ) $
 +
in equi-affine space:
  
<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/a/a010/a010990/a01099030.png" /></td> </tr></table>
+
$$
 +
g _ {ij}  =
 +
\frac{a _ {ij} }{| a |  ^ {1/4} }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099034.png" />. The vector
+
where $  a _ {ij} = ( r _ {1} , r _ {2} , r _ {ij} ) $,
 +
a = \mathop{\rm det} ( a _ {ij} ) $,  
 +
$  r _ {i} = \partial  _ {i} \mathbf r $,  
 +
$  r _ {ij} = \partial  _ {ij} \mathbf r $.  
 +
The vector
  
<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/a/a010/a010990/a01099035.png" /></td> </tr></table>
+
$$
 +
\mathbf N  =
 +
\frac{1}{2}
 +
g  ^ {ks} \nabla _ {k} r _ {s} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099036.png" /> is the symbol of the covariant derivative with respect to the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099037.png" />, determines the direction of the affine normal to the surface. The affine normal passes through the centre of the osculating Lie quadric. The derivational equations
+
where $  \nabla _ {k} $
 +
is the symbol of the covariant derivative with respect to the metric tensor $  g _ {ij }  $,  
 +
determines the direction of the affine normal to the surface. The affine normal passes through the centre of the osculating Lie quadric. The derivational equations
  
<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/a/a010/a010990/a01099038.png" /></td> </tr></table>
+
$$
 +
\partial  _ {j} r _ {i}  = {\Gamma ^ { 1 }  } _ {ij}  ^ {s} r _ {s} +
 +
g _ {ij} N
 +
$$
  
define an intrinsic connection of the first kind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099039.png" /> of the surface. There also arises at the same time an intrinsic connection of the second kind <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099040.png" />, defined by the derivational equations
+
define an intrinsic connection of the first kind $  {\Gamma ^ { 1 }  } _ {ij}  ^ {k} $
 +
of the surface. There also arises at the same time an intrinsic connection of the second kind $  {\Gamma ^ { 2 }  } _ {ij}  ^ {k} $,  
 +
defined by the derivational equations
  
<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/a/a010/a010990/a01099041.png" /></td> </tr></table>
+
$$
 +
\partial  _ {j} \nu _ {i}  = {\Gamma ^ { 2 }  } _ {ij}  ^ {s} \nu _ {s} + A _ {ij} bold \nu ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099042.png" /> is a covariant vector defining the tangent plane to the surface and subject to the normalization condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099043.png" />. The connections
+
where $  bold \nu $
 +
is a covariant vector defining the tangent plane to the surface and subject to the normalization condition $  \mathbf N bold \nu = 1 $.  
 +
The connections
  
<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/a/a010/a010990/a01099044.png" /></td> </tr></table>
+
$$
 +
{\Gamma ^ { 1 }  } _ {ij}  ^ {k} \  \textrm{ and } \  {\Gamma ^ { 2 }  } _ {ij}  ^ {k}
 +
$$
  
are conjugate with respect to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010990/a01099045.png" /> in the sense of A.P. Norden [[#References|[3]]]. The tensor
+
are conjugate with respect to the tensor $  g _ {ij }  $
 +
in the sense of A.P. Norden [[#References|[3]]]. The tensor
  
<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/a/a010/a010990/a01099046.png" /></td> </tr></table>
+
$$
 +
T _ {ij}  ^ {k}  =
 +
\frac{1}{2}
 +
\left ( {\Gamma ^ { 1 }  } _ {ij}  ^ {k} -
 +
{\Gamma ^ { 2 }  } _ {ij}  ^ {k} \right ) ,
 +
$$
  
 
which also plays a major part in [[Projective differential geometry|projective differential geometry]], makes it possible to construct the symmetric covariant tensor
 
which also plays a major part in [[Projective differential geometry|projective differential geometry]], makes it possible to construct the symmetric covariant tensor
  
<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/a/a010/a010990/a01099047.png" /></td> </tr></table>
+
$$
 +
T _ {ijk}  = g _ {ks} T _ {ij}  ^ {s} .
 +
$$
  
 
The two principal surface forms are also constructed: the quadratic form
 
The two principal surface forms are also constructed: the quadratic 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/a/a010/a010990/a01099048.png" /></td> </tr></table>
+
$$
 +
\phi  = g _ {ij}  du  ^ {i}  du  ^ {j}
 +
$$
  
 
and the Fubini–Pick cubic form
 
and the Fubini–Pick cubic 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/a/a010/a010990/a01099049.png" /></td> </tr></table>
+
$$
 +
\psi  = T _ {ijk}  du  ^ {i}  du  ^ {j}  du  ^ {k} .
 +
$$
  
 
These forms are connected by the apolarity condition
 
These forms are connected by the apolarity condition
  
<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/a/a010/a010990/a01099050.png" /></td> </tr></table>
+
$$
 +
g  ^ {ij} T _ {ijk}  = 0 .
 +
$$
  
 
Two such forms, which satisfy supplementary differential conditions, determine the surface up to equi-affine transformations. All these statements have appropriate generalizations in the multi-dimensional case.
 
Two such forms, which satisfy supplementary differential conditions, determine the surface up to equi-affine transformations. All these statements have appropriate generalizations in the multi-dimensional case.
Line 77: Line 176:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''2''' , Springer  (1923)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Salkowski,  "Affine Differentialgeometrie" , de Gruyter  (1934)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.P. Norden,  "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.F. Laptev,  "Differential geometry of multi-dimensional surfaces"  ''Itogi Nauk. Geom. 1963''  (1965)  pp. 3–64  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.A. Shirokov,  A.P. Shirokov,  "Differentialgeometrie" , Teubner  (1962)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,  "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , '''2''' , Springer  (1923)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Salkowski,  "Affine Differentialgeometrie" , de Gruyter  (1934)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.P. Norden,  "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.F. Laptev,  "Differential geometry of multi-dimensional surfaces"  ''Itogi Nauk. Geom. 1963''  (1965)  pp. 3–64  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.A. Shirokov,  A.P. Shirokov,  "Differentialgeometrie" , Teubner  (1962)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 16:09, 1 April 2020


The branch of geometry dealing with the differential-geometric properties of curves and surfaces that are invariant under transformations of the affine group or its subgroups. The differential geometry of equi-affine space has been most thoroughly studied.

In an equi-affine plane any two vectors $ \mathbf a , \mathbf b $ have an invariant $ ( \mathbf a , \mathbf b ) $— the surface area of the parallelogram constructed on $ \mathbf a $ and $ \mathbf b $. With the aid of this concept, the invariant parameter

$$ s = \int\limits _ { t } _ {0} ^ { t } | ( \dot{\mathbf r} , \dot{\mathbf r} dot ) | ^ {1/3} dt, $$

known as the equi-affine arc length, can be constructed for a non-rectilinear curve $ \mathbf r = \mathbf r (t) $. The differential invariant

$$ k = \left ( \frac{d ^ {2} \mathbf r }{ds ^ {2} } ,\ \frac{d ^ {3} \mathbf r }{d s ^ {3} } \right ) $$

is called the equi-affine curvature of the plane curve. Constant equi-affine curvature characterizes curves of the second order. A natural equation $ k = f(s) $ determines a curve up to an equi-affine transformation. The vector $ n = {d ^ {2} \mathbf r } / {d s ^ {2} } $ is directed along the affine normal to a plane curve; the affine normal at a point $ M $, $ k \neq 0 $, is the tangent to the locus of the mid-chords of the curve parallel to the tangent at $ M $, and coincides with the diameter of the parabola which has third-order contact with the curve at $ M $.

Passing to the general affine group, two more invariants of the curve are considered: the affine arc length $ \sigma $ and the affine curvature $ \kappa $. They can be expressed in terms of the invariants $ s $ and $ k $ introduced above:

$$ \sigma = \int\limits k ^ {1/2} ds,\ \ \kappa = \frac{1}{k ^ {3/2} } \cdot \frac{dk}{ds} . $$

(In equi-affine geometry, the magnitudes $ s $ and $ k $ themselves are called the affine arc length and the affine curvature, for the sake of brevity.) The centro-affine arc length, centro-affine curvature, equi-centro-affine arc length and equi-centro-affine curvature of a plane curve are constructed in a similar manner.

In equi-affine space it is possible to assign to any three vectors $ \mathbf a , \mathbf b , \mathbf c $ the invariant $ ( \mathbf a , \mathbf b , \mathbf c ) $, which is the volume of the oriented parallelepiped defined by these vectors. The natural parameter (equi-affine arc length) of a curve $ \mathbf r = \mathbf r (t) $( $ \mathbf r \in C ^ {3} $) is defined by the formula

$$ s = \int\limits _ { t } _ {0} ^ { t } | ( \dot{\mathbf r} , \dot{\mathbf r} dot , \mathbf r ^ { {. } . . } ) | ^ {1/6} dt. $$

The differential invariants $ \kappa = ( \mathbf r ^ \prime , \mathbf r ^ {\prime\prime\prime} , \mathbf r ^ {\prime\prime\prime\prime} ) $, $ \tau = - ( \mathbf r ^ {\prime\prime} , \mathbf r ^ {\prime\prime\prime} , \mathbf r ^ {\prime\prime\prime\prime} ) $, where the primes denote differentiation with respect to the natural parameter, are called, respectively, the equi-affine curvature and the equi-affine torsion of the spatial curve. The study of the curve is reduced to selecting some moving frame; the frame formed by the vectors

$$ \left \{ \dot{\mathbf r} , \dot{\mathbf r} dot , \mathbf r ^ { {. } . . } + \kappa \mathbf r \frac{dot}{4} \right \} $$

and defined by the fourth-order differential neighbourhood of the curve being studied, is especially important. The centro-affine theory of spatial curves has been developed [5].

The following tensor is constructed for a non-developable surface $ \mathbf r = \mathbf r (u ^ {1} , u ^ {2} ) $ in equi-affine space:

$$ g _ {ij} = \frac{a _ {ij} }{| a | ^ {1/4} } , $$

where $ a _ {ij} = ( r _ {1} , r _ {2} , r _ {ij} ) $, $ a = \mathop{\rm det} ( a _ {ij} ) $, $ r _ {i} = \partial _ {i} \mathbf r $, $ r _ {ij} = \partial _ {ij} \mathbf r $. The vector

$$ \mathbf N = \frac{1}{2} g ^ {ks} \nabla _ {k} r _ {s} , $$

where $ \nabla _ {k} $ is the symbol of the covariant derivative with respect to the metric tensor $ g _ {ij } $, determines the direction of the affine normal to the surface. The affine normal passes through the centre of the osculating Lie quadric. The derivational equations

$$ \partial _ {j} r _ {i} = {\Gamma ^ { 1 } } _ {ij} ^ {s} r _ {s} + g _ {ij} N $$

define an intrinsic connection of the first kind $ {\Gamma ^ { 1 } } _ {ij} ^ {k} $ of the surface. There also arises at the same time an intrinsic connection of the second kind $ {\Gamma ^ { 2 } } _ {ij} ^ {k} $, defined by the derivational equations

$$ \partial _ {j} \nu _ {i} = {\Gamma ^ { 2 } } _ {ij} ^ {s} \nu _ {s} + A _ {ij} bold \nu , $$

where $ bold \nu $ is a covariant vector defining the tangent plane to the surface and subject to the normalization condition $ \mathbf N bold \nu = 1 $. The connections

$$ {\Gamma ^ { 1 } } _ {ij} ^ {k} \ \textrm{ and } \ {\Gamma ^ { 2 } } _ {ij} ^ {k} $$

are conjugate with respect to the tensor $ g _ {ij } $ in the sense of A.P. Norden [3]. The tensor

$$ T _ {ij} ^ {k} = \frac{1}{2} \left ( {\Gamma ^ { 1 } } _ {ij} ^ {k} - {\Gamma ^ { 2 } } _ {ij} ^ {k} \right ) , $$

which also plays a major part in projective differential geometry, makes it possible to construct the symmetric covariant tensor

$$ T _ {ijk} = g _ {ks} T _ {ij} ^ {s} . $$

The two principal surface forms are also constructed: the quadratic form

$$ \phi = g _ {ij} du ^ {i} du ^ {j} $$

and the Fubini–Pick cubic form

$$ \psi = T _ {ijk} du ^ {i} du ^ {j} du ^ {k} . $$

These forms are connected by the apolarity condition

$$ g ^ {ij} T _ {ijk} = 0 . $$

Two such forms, which satisfy supplementary differential conditions, determine the surface up to equi-affine transformations. All these statements have appropriate generalizations in the multi-dimensional case.

Many specific classes of surfaces are distinguished in affine and equi-affine spaces: affine spheres (for which the affine normals form a bundle), affine surfaces of revolution (the affine normals intersect one proper or improper straight line), affine minimal surfaces, etc.

In addition to curves and surfaces, other geometrical objects in equi-affine space are also studied, such as congruences and complexes of straight lines, vector fields, etc.

In parallel with equi-affine differential geometry, development is also in progress of the differential geometry of the general affine group and of its other subgroups both in three-dimensional and in multi-dimensional spaces (centro-affine, equi-centro-affine, affine-symplectic, bi-affine, etc.).

References

[1] W. Blaschke, "Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie. Affine Differentialgeometrie" , 2 , Springer (1923)
[2] E. Salkowski, "Affine Differentialgeometrie" , de Gruyter (1934)
[3] A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)
[4] G.F. Laptev, "Differential geometry of multi-dimensional surfaces" Itogi Nauk. Geom. 1963 (1965) pp. 3–64 (In Russian)
[5] P.A. Shirokov, A.P. Shirokov, "Differentialgeometrie" , Teubner (1962) (Translated from Russian)

Comments

For the development of affine differential geometry after W. Blaschke, see [a1].

References

[a1] U. Simon, "Zur Entwicklung der affine Differentialgeometrie nach Blaschke" , Wilhelm Blaschke gesammelte Werke , 4 , Thales Verlag (1985) pp. 35–88
How to Cite This Entry:
Affine differential geometry. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_differential_geometry&oldid=17614
This article was adapted from an original article by A.P. Shirokov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article