Abstract
For a given nondegenerate hypersurfaceM n in affine space ℝn+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 ℝn+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.
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Research Assistant of the National Fund for Scientific Research (Belgium).
The work of the second author was done while he was visiting professor at Katholieke Universiteit Leuven. He would like to express his thanks for their hospitality.
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Dillen, F., Nomizu, K. & Vranken, L. Conjugate connections and Radon's theorem in affine differential geometry. Monatshefte für Mathematik 109, 221–235 (1990). https://doi.org/10.1007/BF01297762
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DOI: https://doi.org/10.1007/BF01297762