Homogeneous spaces of Iwasawa type and rank one
by
Maria J. Druetta
FaMAF, Universidad Nacional de Cordoba, ARGENTINA

Oral Communication

A riemannian manifold M is called a P-space if for every geodesic c in M the associated Jacobi operator R_c is diagonalizable by a parallel orthonormal basis along c, and M is called a C-space if the eigenvalues of the operators R_c are constant along the geodesics. Both classes generalize Riemannian symmetric spaces.

We study the above geometric conditions on homogeneous spaces of Iwasawa type and algebraic rank one. This class, obtained as Lie group of Iwasawa type, contains as a subclass the Damek-Ricci spaces and more generally the irreducible, non flat homogeneous Einstein spaces of nonpositive curvature and algebraic rank one.

A homogeneous space of Iwasawa type and algebraic rank one is a simply connected Lie group associated to a solvable metric Lie algebra s, whose commutator n is 1-dimensional, and a unit vector H orthogonal to n can be chosen so that the restriction of ad_H to n is symmetric with positive eigenvalues.
We obtain, in general, the expressions of the eigenvalues of the Jacobi operators along geodesics associated to eigenvalues of the restriction of ad_H to n. The geometric hypothesis about the Jacobi operators gives restrictions on the Lie algebra s: the eigenvalues of ad_H are 1 or 1/2 when restricted to the center or its orthogonal complement in n, respectively. In the case of Einstein spaces, it follows that the associated Lie group S is a Damek-Ricci space. By using results of [Berndt, Tricerri, Vanhecke, Lecture Notes in Math. 1598] we have, under the Einstein condition, that if S is a P-space of Iwasawa type and algebraic rank one, then it is a symmetric space of noncompact type and rank one. In particular, if M is an irreducible, non flat homogeneous Einstein P- space of nonpositive curvature and algebraic rank one then it is a rank one symmetric space of noncompact type.

Since homogeneous C-spaces of Iwasawa type are Einstein it follows, by using the expression of the eigenvalues of the Jacobi operators along geodesics, that irreducible, non flat homogeneous C-spaces of nonpositive curvature and algebraic rank one are symmetric spaces of noncompact type and rank one.

Date received: May 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-66.