Affine differential geometry

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