CONJUGATE CONNECTIONS AND RADON THEOREM IN AFFINE DIFFERENTIAL GEOMETRY

For a given nondegenerate hypersurface Mn in affine space ℝn+1 there exist an affine connection ∇, called the induced connection, and a nondegenerate metric h, called the affine metric, which are uniquely determined. The cubic form C=∇h is totally symmetric and satisfies the so-called apolarity condition relative to h. A natural question is, conversely, given an affine connection ∇ and a nondegenerate metric h on a differentiable manifold Mn such that ∇h is totally symmetric and satisfies the apolarity condition relative to h, can Mn be locally immersed in ℝn+1 in such a way that (∇, h) is realized as the induced structure? In 1918 J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the case n=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in cases n=2 and n≥3 in terms of the curvature tensor R of the connection ∇. We also provide another formulation valid for all dimensions n: A necessary and sufficient condition for the realizability of (∇, h) is that the conjugate connection of ∇ relative to h is projectively flat. © 1990 Springer-Verlag.