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 have an invariant — the surface area of the parallelogram constructed on and . With the aid of this concept, the invariant parameter

known as the equi-affine arc length, can be constructed for a non-rectilinear curve . The differential invariant

is called the equi-affine curvature of the plane curve. Constant equi-affine curvature characterizes curves of the second order. A natural equation determines a curve up to an equi-affine transformation. The vector is directed along the affine normal to a plane curve; the affine normal at a point , , is the tangent to the locus of the mid-chords of the curve parallel to the tangent at , and coincides with the diameter of the parabola which has third-order contact with the curve at .

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

(In equi-affine geometry, the magnitudes and 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 the invariant , which is the volume of the oriented parallelepiped defined by these vectors. The natural parameter (equi-affine arc length) of a curve () is defined by the formula

The differential invariants , , 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

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 in equi-affine space:

where , , , . The vector

where is the symbol of the covariant derivative with respect to the metric tensor , 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

define an intrinsic connection of the first kind of the surface. There also arises at the same time an intrinsic connection of the second kind , defined by the derivational equations

where is a covariant vector defining the tangent plane to the surface and subject to the normalization condition . The connections

are conjugate with respect to the tensor in the sense of A.P. Norden [3]. The tensor

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

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

and the Fubini–Pick cubic form

These forms are connected by the apolarity condition

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