Conjugate connections and Radon's theorem in affine differential geometry

Abstract

For a given nondegenerate hypersurfaceM n in affine space ℝ n+1 there exist an affine connection ∇, called the induced connection, and a nondegenerate metrich, called the affine metric, which are uniquely determined. The cubic formC=∇h is totally symmetric and satisfies the so-called apolarity condition relative toh. A natural question is, conversely, given an affine connection ∇ and a nondegenerate metrich on a differentiable manifoldM n such that ∇h is totally symmetric and satisfies the apolarity condition relative toh, canM n be locally immersed in ℝ n+1 in such a way that (∇,h) is realized as the induced structure? In 1918J. Radon gave a necessary and sufficient condition (somewhat complicated) for the problem in the casen=2. The purpose of the present paper is to give a necessary and sufficient condition for the problem in casesn=2 andn≥3 in terms of the curvature tensorR of the connection ∇. We also provide another formulation valid for all dimensionsn: A necessary and sufficient condition for the realizability of (∇,h) is that the conjugate connection of ∇ relative toh is projectively flat.