# A CHARACTERIZATION OF SPACE FORMS

• Kim, Dong-Soo (Department of Mathematics, Chonnam National University) ;
• Kim, Young-Ho (Department of Mathematics Education, Teachers College, Kyungpook National University)
• Published : 1998.11.01

#### Abstract

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 $\ge{n+1}$, then $M^n$ is a Riemannian space form.