Convex hypersurfaces with transnormal horizons are spheres


Let M be a smooth (=C^∈fty), compact, connected hypersurface of Euclidean (n+1)-space R<sup>n+1</sup>,n≥ 2, with nowhere-zero Gaussian curvature. Thus M is differeomorphic to the n-sphere S<sup>n</sup> and every affine tangent hyperplane meets M in just one point.Let λ be any (straight) line in R<sup>n+1</sup> and let M_λ denote the set of points of M at which the tangent hyperplane is parallel to λ.We call M_λ the λ-horizon of M. If, for every λ, M_λ is a transnormal submanifold of R<sup>n+1</sup> [5] we shall say that M is horizon-transnormal.In this paper we show that if M is horizon-transnormal then M is a round sphere.The converse is obviously true.We show in §2 that if M is horizon-transnormal then it is transnormal.If M is transnormal then every λ-outline ω_(Error rendering LaTeX formula)M_λ$ is contained in a hyperplane normal to λ.It is then a consequence of a classical result that M must be an n-ellipsoid. Consequently, due to its transnormality, M is a round n-sphere.

DOI Code: 10.1285/i15900932v7n2p167

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