Conjugate connections and Radon's theorem in affine differential geometry (Q748908)

For a given nondegenerate hypersurface \(M^ n\) in affine space \(R^{n+1}\) there exist an affine connection \(\nabla\), called the induced connection, and a nondegenerate metric h, called the affine metric, which are uniquely determined. The cubic form \(C=\nabla h\) is totally symmetric and satisfies the so-called apolarity condition relative to h. A natural question is, conversely, given an affine connection \(\nabla\) and a nondegenerate metric h on a differentiable manifold \(M^ n\) such that \(\nabla h\) is totally symmetric and satisfies the apolarity condition relative to h, can \(M^ n\) be locally immersed in \(R^{n+1}\) in such a way that (\(\nabla.h)\) 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 \(n\geq 3\) in terms of the curvature tensor R of the connection \(\nabla\). For \(n=2\), the condition is that \(trace_ h\{(X,W)\to (\nabla_ XR)(Y,Z)W\}=0\) for all tangent vectors Y and Z at every point. For \(n\geq 3\), the condition is that \(R(X,Y)Z=0\) whenever X, Y, and Z are perpendicular relative to h. We also provide another formulation valid for all dimensions n: A necessary and sufficient condition for the realizability of (\(\nabla,h)\) is that the conjugate connection of \(\nabla\) relative to h is projectively flat.