Eine Kennzeichnung der projektiven Unterräume in dem komplexen projektiven Raum nach ihren Geodätischen

Ohne Zusammenfassung     

Characteristic roots and vectors of a diifferentiable family of symmetric matrices

LetA(x) be a differentiable family of k × k symmetric matrices where x runs through a domain D in RⁿWe prove that if λ is a continuous function onDsuch that, for every x εD,λ(x) is a characteristic root of A(x) of constant multiplicity m, then λ is a differentiable function and there exists, locally, a differentiable family of ortho-normal bases for the eigenspace. The case n = 1 has been known in the standard treatises on the perturbation theory for linear operators.     

Conjugate connections and Radon's theorem in affine differential geometry

For a given nondegenerate hypersurfaceM ⁿ in affine space ? 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 ? 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.