A geometric proof of the existence of the Green bundles

Abstract

We give a new proof of the existence of the Green bundles. Let (M, g) be a compact Riemannian manifold, and denote by 9t its geodesic flow on the tangent bundle TM. Let -r : TM -* M be the canonical projection and for all 0 E TM let V(0) be the kernel of d-ro. Two points 01, 02 are said to be conjugate if 02 g9tO and dgtV(01) n V(02) & o0 It was proved by Hopf [5] that a two-dimensional torus without conjugate points is flat. Afterwards, Green [8] proved that the integral of the scalar curvature of a manifold without conjugate points is nonpositive and it vanishes if the metric is locally flat. A main ingredient was the existence, under the condition of no conjugate points, of the following bundles: (1) Es(0) lim dg_tV(gt(0)), t-> oo (2) E'(0) = lim dgtV(g_t(0)). t-* oo Hopf's result was generalized to higher dimensions in [2], but there are still new rigidity type results using these bundles; see for example [1]. These bundles have other applications: among other ideas they where used by Freire and Mane' [7] to obtain estimates of the topological entropy. Foulon [6] generalized this result to the case of Finsler metrics. The bundles were also used by Eberlain [4] who proved that these are transverse if and only if the geodesic flow is Anosov. This result was also generalized to the case of convex Hamiltonians without conjugate points; see [31. The purpose of this note is to give a new proof of the following Theorem. If the geodesic flow 9t of a compact manifold does not have conjugate points, then for every 0 in TM the limits (1) and (2) exist. We recall from [4] the definition of the connection map K: T0TM -> T7r(o)M. For ( on T0TM let Z: (-c, e) -* TM be a curve with initial velocity (. Define K(s) = Z'(O) to be the covariant derivative of Z along the curve ir o Z. The definition does not depend on the curve Z. Received by the editors February 8, 2001. 2000 Mathematics Subject Classification. Primary 37D40. The author was partially supported by CONACYT-Mexico grant #28489-E and EPSRCUnited Kingdom GR/M5610. ?)2002 American Mathematical Society