Does “Δ 2 d 2-n = 0 on a Riemannian manifold” imply flatness?

M. Berger, P. Gauduchon andE. Mazet,Le spectre d'une variété riemannienne, Springer, Berlin, 1971.MR 43: 8025 A. L. Besse,Manifolds all of whose geodesics are closed, Springer, Berlin, 1978.MR 80c: 53044 R. Caddeo, Riemannian manifolds on which the distance function is biharmonic,Rend. Sem. Mat. Univ. e Politec. Torino 40 (1982), 93–101.MR 84i: 53040 R. Caddeo andP. Matzeu, Riemannian manifolds satisfyingΔ 2 r k = 0. (To appear) P. Carpenter, A. Gray andT. J. Willmore, The curvature of Einstein symmetric spaces,Quart. J. Math. Oxford 33 (1982), 45–64.MR 84k: 53048 B. Y. Chen andL. Vanhecke, Differential geometry of geodesic spheres,J. Reine Angew. Math. 325 (1981), 28–67.MR 82m: 53038 A. Gray andM. Pinsky, The mean exit time from a small geodesic ball in a Riemannian manifold,Bull. Sci. Math. (To appear) A. Gray andL. Vanhecke, Riemannian geometry as determined by the volumes of small geodesic balls,Acta Math. 142 (1979), 157–198.MR 81i: 53038 S. Helgason,Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978.MR 80k: 53081 O. Loos,Symmetric spaces I–II, W. A. Benjamin, New York, 1969.MR 39: 365 H. Ruse, A. G. Walker andT. J. Willmore,Harmonic spaces, Cremonese, Roma, 1961.MR 25: 5456