Jacobi fields and osculating rank of the Jacobi operator in some special classes of homogeneous Riemannian spaces

Abstract

The geometry of Riemannian symmetric spaces is really richer than that of Riemannian homogeneous spaces. Nevertheless, there exists a large literature of special classes of homogeneous Riemannian manifolds with an important list of features which are typical for a Riemannian symmetric space. Normal homogeneous spaces, naturally reductive homogeneous spaces or g. o. spaces are some interesting examples of these classes of spaces where, in particular, the Jacobi equation can be also written as a differential equation with constant coefficients and the osculating rank of the Jacobi operator is constant. Compact rank one symmetric spaces are among the very few manifolds that are known to admit metrics with positive sectional curvature. In fact, there exist only three non-symmetric (simply-connected) normal homogeneous spaces with positive curvature: V 1 = S p (2)/SU(2), V 2 = SU(5)/(S p (2) × S 1 ), given by M. Berger and the Wilking’s example V 3 = (SU(3) SO(3))/U • (2). Here, we show some geometric properties of all these spaces, properties related with the existence of isotropic Jacobi fields and the determination of the constant osculating rank of the Jacobi operator. It provides different way to ”measure” of how they are so close or not to the class of compact rank one symmetric spaces.