Maria J. Druetta

Negatively curved homogeneous Osserman spaces

Abstract We show that solvable Lie groups of Iwasawa type satisfying the Osserman condition are symmetric spaces of noncompact type and rank one. As a consequence, the Osserman conjecture holds for non-flat homogeneous manifolds of nonpositive curvature.

On harmonic and 2-stein spaces of Iwasawa type

Abstract We study geometric properties of solvable metric Lie groups S of Iwasawa type; in particular harmonicity and the 2-stein condition. One restriction we obtain is that harmonic spaces of Iwasawa type have algebraic rank one, that is, the commutator subgroup of S has codimension one. We show that among Carnot solvmanifolds the only harmonic spaces are the Damek–Ricci spaces. Moreover, this rigidity result remains valid if harmonicity is replaced by the weaker 2-stein condition. As an application, we show that a harmonic Lie group of Iwasawa type with nonsingular 2-step nilpotent commutator subgroup is, up to scaling, a Damek–Ricci space.

The Lie algebra of isometries of homogeneous spaces of nonpositive curvature

LetM be a simply connected, homogeneous space of nonpositive curvature, and letG be the isometry group ofM. In this paper we study the Lie algebra ofG; we describe the set of roots with respect to a particular abelian subalgebra, and its stability subalgebra at a special point ofM (∞). These descriptions are then used to give conditions which are equivalent to the fact thatM is a symmetric space of noncompact type; we also obtain a new criterion for the visibility property ofM, in terms of the action ofG onM (∞).

D’Atri Spaces of Type k and Related Classes of Geometries Concerning Jacobi Operators

In this article, we continue the study of the geometry of k-D’Atri spaces, 1≤k ≤n−1 (n denotes the dimension of the manifold), begun by the second author. It is known that k-D’Atri spaces, k≥1, are related to properties of Jacobi operators Rv along geodesics, since she has shown that

Clifford translations in manifolds without focal points

{\operatorname{tr}}R_{v}

Geometry of D’Atri spaces of type k

,

D'Atri spaces of Iwasawa type

{\operatorname{tr}}R_{v}^{2}

The relationship between D'Atri and

are invariant under the geodesic flow for any unit tangent vector v. Here, assuming that the Riemannian manifold is a D’Atri space, we prove in our main result that

k

{ \operatorname{tr}}R_{v}^{3}

-D'Atri spaces

is also invariant under the geodesic flow if k≥3. In addition, other properties of Jacobi operators related to the Ledger conditions are obtained, and they are used to give applications to Iwasawa type spaces. In the class of D’Atri spaces of Iwasawa type, we show two different characterizations of the symmetric spaces of noncompact type: they are exactly the