Damek-Ricci spaces satisfying the Osserman-\(p\) condition (Q2767716)

Let \(M^n\) be a Riemannian manifold, \(R\) its curvature tensor and \(R_X\) the Jacobi operator defined as \(R_XY=R(Y, X)X\). One says that \(M\) satisfies the Osserman-\(p\) condition (Oss-\(p\), for short), if for any set of \(p\) orthonormal vectors \(E=\{X_1, \ldots, X_p\}\), the symmetric operator \(J_E=\sum^p_{j=1}R_{X_j}\) has constant eigenvalues.NEWLINENEWLINENEWLINEP. Gilkey posed the question whether a nonpositively curved homogeneous manifold \(M^n\) satisfying the Oss-\(p\) condition must have constant curvature. The present paper gives an affirmative answer to Gilkey's question in a subclass of the Damek-Ricci spaces. More precisly:NEWLINENEWLINENEWLINELet \(N\) be a H-type two-step nilpotent Lie group, \(S=R^+N\) the semidirect product of the multiplicative group \(R^+\) \(N_0\) \(S\) endowed with a suitable left invariant metric is called the Damek-Ricci space. This class of spaces includes the noncompact symmetric spaces of rank one as particular cases.NEWLINENEWLINENEWLINEUsing the properties of the curvature tensors and the Jacobi operators on Damek-Ricci spaces, the authors prove that if \(M\) is a Damek-Ricci space of dimension \(n\) satisfying the Oss-\(p\) condition for \(1<p<n\), then \(M\) is the real hyperbolic \(n\)-space.