Finite dimensional imbeddings of harmonic spaces

Abstract

In a noncompact harmonic manifold $M$ we establish finite dimensionality of the eigenspaces $V_λ$ generated by radial eigenfunctions of the form $\cosh r + c$. As a consequence, for such harmonic manifolds, we give an isometric imbedding of $M$ into $(V_λ,B)$, where $B$ is a nondegenerate symmetric bilinear indefinite form on $V_λ$ (analogous to the imbedding of the real hyperbolic space $I\!\!\!H^{n}$ into $I\!\!\!R^{n+1}$ with the indefinite form $Q(x,x) = -x_0^2 + \sum x_i^2$). Finally we give certain conditions under which $M$ is symmetric.