Homogeneous Riemannian manifolds and the visibility axiom
In this paper we study the Riemannian homogeneous spaces H of nonpositive sectional curvature satisfying the visibility axiom. If G is a simply transitive and solvable Lie group of isometries of H, the action of G in the points at infinity H(\(\infty)\) of H is completely described even when H has no focal points. We also describe the Lie groups that equipped with a left invariant metric of nonpositive curvature, satisfy the visibility axiom. As a consequence, we obtain a characterization of the Lie groups admitting left invariant metrics satisfying the visibility axiom. They are the same as those admitting left invariant metrics of negative curvature. Moreover, homogeneous visibility manifolds do not necessarily have negative curvature. We exhibit three-dimensional homogeneous visibility manifolds with some zero sectional curvature. Finally, a relationship between the algebraic structure of G and the geometric structure of H is obtained if visibility fails.
scientific article; zbMATH DE number 3897771