Let M be a complete Riemannian manifold. M has no conjugate points if in its universal covering space any two points can be joined by a unique geodesic. If the sectional curvature is nonpositive, then there are no conjugate points. The converse is not true. However a manifold without conjugate points enjoys many interesting properties of nonpositively curved manifold. For instance, E. Hopf proved that a 2-dimensional torus T2 without conjugate points is flat. It was proved by Avez [1] that this is still true for higher dimensional torus under the stronger assumption of no focal points. However it is still an open question for the general case of no conjugate points. In connection of this problem, and also on its own interest, the asymptotic behavior of geodesic rays in a simply connected complete manifold without conjugate points has been studied [8], [9], [10]. The asymptotic behavior of a ray is related to the so called Busemann function, which is in a sense a limit function of the distance functions from the points on the ray. The Busemann function is C'-differentiable [8] and in the case of nonpositive curvature it is C2-differentiable [5], [13]. The most important tool in the study of these problems has been stable Jacobi field, which ; is a solution of the Jacobi equation satisfying some boundary conditions. We will show that i the Busemann function is C2-differentiable if the stable Jacobi tensor has a certain continuity property. There are examples of manifolds without conjugate points which do not have this continuity [2]. This example however does not rule out the possibility of C2-differentiability without the continuity of the stable Jacobi tensor. It is not yet clear what can be the optimal condition for the differentiability of the Busemann function.
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