Finite dimensional imbeddings of harmonic spaces

TL;DR

This paper proves that harmonic manifolds with certain radial eigenfunctions have finite-dimensional eigenspaces, enabling an isometric embedding into a space with an indefinite bilinear form, and explores conditions for the manifold's symmetry.

Contribution

It establishes finite dimensionality of eigenspaces in harmonic manifolds and constructs an isometric embedding into an indefinite bilinear space, extending hyperbolic space embeddings.

Findings

Finite dimensionality of eigenspaces $V_{\lambda}$ for harmonic manifolds.

Construction of an isometric embedding into a space with an indefinite form.

Conditions identified under which the harmonic manifold is symmetric.

Abstract

In a noncompact harmonic manifold $M$ we establish finite dimensionality of the eigenspaces $V_{\lambda}$ 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 $(V_{\lambda},B)$, where $B$ is a nondegenerate symmetric bilinear indefinite form on $V_{\lambda}$ (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.