Tangent sphere bundles with constant trace of the Jacobi operator

Abstract

We consider the unit tangent sphere bundle T1M of a Riemannian manifold (M, g) equipped with the canonical contact metric structure \({(\eta, \xi, \phi, \bar{g})}\) . We study the geometric properties of the base manifold M under the assumption that the trace of the Jacobi operator with respect to the characteristic vector field ξ is constant. Moreover, we give examples of Einstein Riemannian manifolds, other than rank one symmetric spaces or globally Ossermann manifolds, the unit tangent sphere bundles of which have constant trace of the Jacobi operator. Finally, we extend our main results to the tangent sphere bundle TrM of constant radius r > 0.