Finite-dimensional imbeddings of harmonic spaces (Q1301562)

Let \((M,g)\) be a Riemannian manifold, \(m\in M\) and \(\omega_m= (\text{det }g_{ij})^{1/2}\) the volume density function with respect to normal coordinates centered at \(m\). Then, \((M,g)\) is said to be harmonic if and only if \(\omega_m\) is a radial function for each \(m\). The only known examples are the \((M,g)\) which are locally isometric to a two-point homogeneous space or to a Damek-Ricci space. By using the Laplacian \(\Delta\) for a compact strongly harmonic space \((M,g)\), it is possible to show that there exists an isometric minimal immersion of \((M,g)\) into the finite-dimensional vector space \(V_\lambda\) of eigenfunctions of \(\Delta\) corresponding to the eigenvalue \(\lambda\) [see \textit{A. L. Besse}, ``Manifolds all of whose geodesics are closed'', Ergebnisse der Math. 93, Springer-Verlag, Berlin (1978; Zbl 0387.53010)]. This has been extended by \textit{Z. I. Szabó} in [J. Differ. Geom. 31, 1-28 (1990; Zbl 0686.53042)], where he showed that a similar minimal isometric immersion into the Hilbert space \(L^2(M)\) exists for arbitrary harmonic manifolds. For compact strongly harmonic spaces Szabó's construction gives Besse's construction. These results play an important role in the proof by Szabó of the Lichnerowicz conjecture (harmonic Riemannian manifolds are locally isometric to a two-point homogeneous space) for compact manifolds with finite fundamental group. In this paper, the authors treat the case of noncompact harmonic manifolds. Let \(V_\lambda\) denote the eigensubspaces generated by radial eigenfunctions of the form \(\text{cosh }r+ c\), \(r\) being the radius. First, they show that \(V_\lambda\) is finite-dimensional. Then, they construct an isometric embedding of \(M\) into \((V_\lambda, B)\) where \(B\) is some nondegenerate symmetric bilinear indefinite form on \(V_\lambda\) and they show that this embedding is minimal in some sphere in \((V_\lambda, -B)\). Furthermore, they show that all the already known examples of noncompact harmonic spaces belong to the considered class. Finally, they also treat some additional conditions under which the harmonic spaces considered here turn out to be locally symmetric.