A GEOMETRIC PROOF OF THE EXISTENCE OF THE GREEN BUNDLES

RENATO ITURRIAGA

Abstract. We give a new proof of the existence of the Green Bundles.

Let (M, g) be a compact Riemannian manifold, denote by g_{t} its geodesic flow on the tangent bundle T M. Let π : T M → M be the canonical projection and for all θ ∈ T M let V (θ) be the kernel of dπ_{θ}. Two points θ_{1}, θ_{2} are said to be conjugated if θ_{2} = g_{t}θ_{1} and dg_{t}V (θ_{1}) ∩ V (θ_{2}) ≠ 0.

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 non positive and it vanishes if the metric is locally flat. A main ingredient was the existing, under the condition of no conjugate points, of the following bundles:

E^{s}(θ) = lim_{t→∞} dg_{−t}V (g_{t}(θ)),  (1)
E^{u}(θ) = lim_{t→∞} dg_{t}V (g_{−t}(θ)).  (2)

Hopf result was generalised 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 Mañé, [7], to obtain estimates of the topological entropy. Foulon, [6], generalised 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 generalised to the case of convex Hamiltonians without conjugate points. See [3]. The purpose of this note is to give a new proof of the following

Date: January 2001.

Mathematics subject classification 2000 37D40.

Partially supported by a CONACYT-México grant # 28489-E and EPSRC-United Kingdom GR/M5610.

Theorem If the geodesic flow g_{t} of a compact manifold does not have conjugate points then for every θ in T M the limits (1) and (2) exist.

Let ω be the canonical symplectic structure on T M. The space T_{θ}T M splits as the direct sum of two Lagrangian subspaces: V (θ) and the horizontal space H(θ), given by the kernel of the connection map.

Denote by E_{T} = dg_{−T} V (g_{T} (θ)), since the geodesic flow preserves the symplectic structure it follows that, E_{T} is a Lagrangian subspace. It is straightforward to verify then, that it is the graph of a symmetric matrix,

S_{T} : T_{π(θ)}M ∼ H(θ) → V (θ) ∼ T_{π(θ)}M.

Recall that it is possible to give a partial order in the set of symmetric matrices defining A ≻ B if A − B is positive definite. To prove the theorem it is enough to see that the sequence S_{T} is monotone and bounded. The original proof of this fact is a slightly clumsy calculation.

For all t < s the matrix S_{s} − S_{t} is symmetric. We claim that the signature does not change in the region contained in ℝ^{2} such that 0 < t < s. Otherwise we can find t_{0} < s_{0} such that S_{s_{0}} − S_{t_{0}} has 0 as an eigenvalue, this implies that θ_{s_{0}} and θ_{t_{0}} are conjugate.

In the other hand the Jacobi equation has the form

ḣ = v  (3)
v̇ = F (h, v)

In particular if s ≪ 1 the solution of (3) is h(s) = h_{0} + sv + O(s^{2}). Hence S_{s} = −\frac{1}{s}I + O(s^{2}), where I is the identity matrix. Consequently S_{s} − S_{t} is positive definite and S_{s} is monotone increasing as s tends to infinity.

Similarly for all t < 0 < s the matrix S_{s} − S_{t} has constant signature. Moreover by the same estimate the matrix S_{s} − S_{t} is negative definite, for |s|, |t| ≪ 1. So S_{s} is bounded by S_{t} for any t < 0. Similar arguments apply for the unstable bundle E^{u}.

Finally let us remark that the same argument, see [3] to define the order of symmetric matrices, applies for convex Hamiltonians without conjugate points.

References

[1] M.L. Bialy Rigidity for periodic magnetic fields Ergodic Th. and Dynam. Sys. 20 (2000) 1619-1626.

[2] D. Burago, S. Ivanov Riemannian Tori without conjugate points are flat GAFA (1994) 259-269.

[3] G. Contreras, R. Iturriaga, Convex Hamiltonians without conjugate points, Ergod. Th and Dynam. Sys. 19 (1999) 901–952.

[4] P. Eberlain When is a geodesic flow af Anosov type? I J. Differential Geometry 8 (1973) 437-463.

[5] E. Hopf Closed surfaces without conjugate points Proc. Natl. Acad. Sci 34(1948) 47-51.

[6] P. Foulon Estimation de l’entropie des systemes lagrangiens sans points conjugués. Ann. Inst. Henri Poincare 57 (1992) 117-146.

[7] A. Freire R. Mañé On the entropy of the geodesic flow of manifolds without conjugate points Inv. Math. 69 (1982) 375-392.

[8] L.W. Green A theorem of E. Hopf Michigan Math.J. 5 (1958) 31-34.

CIMAT,, A.P. 402, 36000, Guanajuato. Gto., México., I.Newton Institute, Cambridge University

 E-mail address: renato@fractal.cimat.mx