Finite dimensional imbeddings of harmonic spaces

Details Finite dimensional imbeddings of harmonic spaces Finite dimensional imbeddings of harmonic spaces

1996-03-23

arxiv.org/abs/dg-ga/9603012v1

Mathematics

Differential Geometry

10 pages, latex (e-mail: kram@..., aranjan@ganit.math.iitb.ernet.in)

Scientific paper

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.

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