Convex hypersurfaces with transnormal horizons are spheres

Abstract

Let M be a smooth , compact, connected hypersurface of Euclidean -space ,, with nowhere-zero Gaussian curvature. Thus M is differeomorphic to the n-sphere and every affine tangent hyperplane meets M in just one point.Let λ be any (straight) line in and let denote the set of points of M at which the tangent hyperplane is parallel to λ.We call the λ-horizon of M. If, for every λ, is a transnormal submanifold of [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.