Caracterizações da esfera em formas espaciais

Abstract

In this work we present three characterizations of the sphere. Initially, it will be shown that given a compact and oriented hypersurface Mⁿ e x: M → Qⁿ⁺¹c a isometric immersion, x(M) is a geodesic sphere in Qⁿ⁺¹c if, and only if, Hr+1 is a nonzero constant and the set of points that are omitted in Qⁿ⁺¹c by the totally geodesic hypersurfaces (Qⁿc )p tangent to x(M) is non-empty. As a second result, let M be an orientable compact and connected hypersurface with non-negative support function of the Euclidean space Rn+1 and Minkowski's integrand . We prove that the mean curvature function of the hypersurface M is the solution of the Poisson equation Δϕ = σ if, and only if, M is isometric to the n-sphere Sⁿ(c) of constant curvature c. similar characterization is proved for a hypersurface with the scalar curvature satisfying the same equation. For the third result we consider an isometric immersion x : M → Qⁿ⁺¹, where M is a compact hypersurface such that x(M) is convex, and it will be proved that if any r-mean curvature is such that Hr ≠ 0 and there are nonnegative constants C1, C2, ..., Cr-1 tais que Hr =∑ⁿ⁻¹(i=1) Ci Hi;; then x(M) is a geodesic sphere, where Qⁿ⁺¹ is Rⁿ⁺¹, Hⁿ⁺¹ or Sⁿ⁺¹+ .