Applications of Jacobi and Riccati Equations along Flows to Riemannian Geometry

Abstract

In the present paper we show a model for geodesic flows on the unit tangent bundles of complete Riemannian manifolds. By treating it as in the study of manifolds without conjugate points we have two theorems of the same type as E. Hopf and L. Green proved. One is for spaces of constant curvature instead of flat manifolds. The other is for differentiable flows without conjugate points, and in particular, gradient flows. In addition, we give the formula of the same type as R. Ossermann and P. Sarnak did. As its application, we get the simpler proof of the extension due to W. Ballmann and W. P. Wojtkowski.