Bulletin of the
Korean Mathematical Society BKMS

For a Riemannian manifold $(M^n, g)$, we consider the space $V(M^n, g)$of all smooth functions on $M^n$ whose Hessian is proportional to the metric tensor $g.$ It is well-known that if $M^n$ is a space form then $V(M^n)$ is of dimension $n+2.$ In this paper, conversely, we prove that if $V(M^n)$ is of dimension $\geq\;\;n+1,$ then $M^n$ is a Riemannian space form.

Keywords: Hessian, warped product, space form

MSC numbers: 53B20, 53C20