Negatively curved homogeneous Osserman spaces
Let \(M\) be a riemannian manifold, \(R\) its curvature tensor and \(R_X\) the curvature operator, defined by \(R_{X}Y = R(Y,X)X\), \(X\) a unit tangent vector. When \(R_X\) has constant eigenvalues, counting multiplicities, independently of \(X\) in \(T_{p}M\) and of the choice of \(p\) in \(M\), the riemannian manifold is said to satisfy the Osserman condition. It is immediate to see that it is satisfied by rank one symmetric spaces. Based on results obtained in R. Osserman, P. Sarnak, A new curvature invariant and entropy of geodesic flows, Inventiones Math, 77 (1984) 455-462, Osserman cojectured that locally rank one symmetric spaces are the only non-flat riemannian manifolds satisfying this condition. The authors proved the validity of the above conjecture in the case of a homogeneous manifold of nonpositive curvature. An outline of the proof is as follows. Every riemannian manifold satisfying the Osserman condition is Einstein, hence by applying results from (J. Heber, Noncompact homogeneous Einstein spaces, Inventiones Math. 133 (1998) 279-352) it is sufficient to verify the conjecture in the case of a solvable Lie groups of Iwasawa type. This is proved in Theorem 3.4: If \(\mathcal S\) is a Lie algebra of Iwasawa type satisfying the Osserman condition, then the associated simply connected Lie group \(S\) with left invariant metric is a symmetric space of noncompact type and rank one.
scientific article; zbMATH DE number 1367713
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