Some differential equations on Riemannian manifolds

Abstract

\S 1. Introduction.Let $(M, g)$ be a Riemannian manifold of dimension $m\geqq 2$ and let $\nabla$ denote the Riemannian connection defined by $g$.In this paper we study the following system of differential equations of order three:(1.1)where $k$ is a positive constant.Originally the differential equations (1.1) come from some study of the Laplacian on a Euclidean sphere ( $S^{m}$ ; k) of constant curvature $k$ .The first eigenvalue of the Laplacian on ( $S^{m}$ ; k) is $mk$ and each eigenfunction $h$ corresponding to $mk$ satisfies the following system of differential equations of order two:(1.2)The second eigenvalue is $2(m+1)k$ and each eigenfunction $f$ corresponding to $2(m+1)k$ satisfies (1.1).Assuming the existence of a non-constant function $h$ satisfying (1.2) on a Riemannian manifold $(M, g)$ many mathematicians studied differential geometric properties of $(M, g)$ (cf.S. Ishihara and Y. Tashiro [11], M. Obata [14], [15], Y. Tashiro [22], etc.).In this case grad $f$ is an infinitesimal conformal trans- formation.Assume that there is a non-constant function $f$ satisfying (1.1) on $(M, g)$ .Then grad $f$ is an infinitesimal projective transformation and is a k-nullity vector field on $(M, g)$ .The converse is also true (cf.Proposition 2.1).This gives a geometric meaning of (1.1).The system of differential equations (1.1) was first studied by M. Obata [15] and he announced the following.THEOREM A. Let $(M, g)$ be a complete and simply connected Riemannian manifold.In order for $(M, g)$ to admit a non-constant function $f$ satisfying (1.1)