Growth functions of geodesic spheres

Abstract

Let M be an analytic Riemannian manifold of dimension n. For small r>0, denote by Sm(r) the (n-1)-dimensional volume of a geodesic sphere Gm(r) of radius r with center m. Then the growth function A(s) of Gm(r) is given by A(s)=Sm(r+s) for small s≥0. We characterize the space of constant curvature by the equations A"(s)+C(s)A(s)=0 for all Gm(r) or A'''(s)+C (s)A''(s)+C (s)A' (s)+C (s)A(s)=0 for all Gm(r).