A characterization of quadrics by the principal curvature functions

Abstract

We consider the difference tensor field of the Levi-Civita connections of the first and third fundamental form for non-degenerate hypersurface immersions in space forms. Within this framework, we characterize quadrics in terms of their principal curvature functions.

A characterization of revolution quadrics by a system of partial differential equations

“…Let M be a C ∞ complete Riemannian manifold of dimension n ≥ 2 with metric h. Denote by ∇ 2 the Hessian matrix of the second covariant derivatives in the metric h. The well known theorem of Obata [9] states that if the system ∇ 2 w + k 2 wh = 0, k = const > 0, (1) admits a solution w ≡ 0 then M is isometric to a sphere of radius 1/k in R n+1 . There have been many important applications of Obata's theorem to problems of isometry and conformality of manifolds with Euclidean spheres [10], [14], [6].…”

“…There have been many important applications of Obata's theorem to problems of isometry and conformality of manifolds with Euclidean spheres [10], [14], [6]. On the other hand, some of the known characterizations of quadrics in Euclidean space use systems of third order partial differential equations (PDE's) defining second order spherical harmonics (see [15] and other references there), while other characterizations are based on affine invariants and affine isoperimetric inequalities [7], [8], or on postulated relations between principal curvatures [1], [4], [5], [15]. The proofs usually require substantial efforts.…”

Yau's Problem on a Characterization of Rotational Ellipsoids

“…The authors of [2] generalized the problem and its solution to hypersurfaces in space forms. Lemma 3.5.1 below shows the close relation of the problems of Yau and Voss, resp.…”