In 1916 Blaschke and his school started to investigate systematically the differential geometry of surfaces in real affine 3-space d~ with respect to the unimodular group.
The fundamental local theorem of this theory reads roughly as follows ([2, Β§52]; [-9, Β§24]): Let M be a 2-dimensional, connected, oriented C ~manifold, G a Riemannian metric and A a totally symmetry (0.3)-tensor field such that the corresponding integrability conditions (cf. [2, Β§60 (155)]) are fu!~lled. Then there exists an immersion x: M --+d 3 such that x(M) is locally strongly convex, G is the equiaffine Berwald-Blaschke metric and A the equiaffine cubic form: x(M) is uniquely determined up to equiaffine transformations of J3.
There are analogous results for hypersurfaces with respect to the unimodular and the centroaffine group, as well as in the so called relative differential geometry (cf. [-2, Β§65]; [6, Β§II.3]; [9, Β§19, 24]).
The tool for the main theorem are the structure equations. In the Euclidean case there are two systems, the Weingarten equations and the Gauss equations, while there are three systems of structure equations in the affine differential geometries: the affine Weingarten equations and two systems corresponding to the Gauss equations, one induced by the hypersurface immersion x, the other one by the conormal immersion X.
The structure equations for x and X define two connections IF and 2F, resp.; the Levi-Civita F connection of the affine metric G is the arithmetic mean:
It was already pointed out by Blaschke ([2, p. 169]) that one system of integrability conditions, the affine Codazzi equations, get a much simpler structure if one uses covariant differentiation with respect to 1F instead with respect to F. This idea was extended by Barthel and some of his students. They developed the calculus of the equiaffine differential geometry with 1F and G as basic geometric quantities and gave another version of the main theorem. This led to generalizations of affine structures to differentiable manifolds (cf. as surveys, e.g., [1] and [15, Β§III.1.3]).
Another version of the fundamental theorem is due to Norden (cf. [9, p. 228]); his theorem is based on the connection 2F and the relative support