View source for 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.

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 [[#References|[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 [[#References|[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|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====
<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====
For the development of affine differential geometry after W. Blaschke, see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U. Simon,   "Zur Entwicklung der affine Differentialgeometrie nach Blaschke" , ''Wilhelm Blaschke gesammelte Werke'' , '''4''' , Thales Verlag  (1985)  pp. 35–88</TD></TR></table>