A note on some remarkable differential equations on a Riemannian manifold

Abstract

The Fischer-Marsden conjecture asserts that an n-dimensional compact manifold admitting a nontrivial solution of the so-called Fischer-Marsden differential equation is necessarily an Einstein space. If this were true, then a classical theorem of Obata would imply that the underlying manifold is either a standard sphere or a Ricci flat space. Although counterexamples to this conjecture have been found by Kobayashi and Lafontaine, it has recently been proved by Cernea and Guan that the Fischer-Marsden conjecture holds, provided that the space of nonconstant solutions of the Fischer-Marsden equation is of dimension at least n, the authors actually proving that in this case (M,g) is nothing but a standard sphere. The main aim of this article is to show that any compact Riemannian manifold of positive Ricci curvature that admits a nontrivial concircular vector field with the potential function satisfying the Fischer-Marsden equation must be isometric to a standard sphere and the converse is also valid. Moreover, we prove that the existence of a nontrivial solution to another remarkable differential equation on Riemannian manifolds, namely the stationary Schrödinger equation, it also leads to a characterization of the sphere, provided that some pinching conditions are satisfied.