Finite dimensional imbeddings of harmonic spaces

For a noncompact harmonic manifoldM we establish finite dimensionality of the eigensubspacesVγ generated by radial eigenfunctions of the form coshr+c. As a consequence, for such harmonic manifolds, we give an isometric imbedding ofM into (Vγ,B), whereB is a nondegenerate symmetric bilinear indefinite form onVγ (analogous to the imbedding of the real hyperbolic spaceHn into ℝn+1 with the indefinite formQ(x,x)=−x02+Σxi2). This imbedding is minimal in a ‘sphere’ in (Vγ,B). Finally we give certain conditions under whichM is symmetric.