Moment Map, a Product Structure, and Riemannian Metrics with no Conjugate Points

Raul M. Aguilar
2005 Communications in analysis and geometry  
1 , · · · ∂ ∂u ± n ± , where T (T M)| O = [T (T M)] + ⊕ [T (T M)] − is the decomposition in the ±1-eigenbundles of P, hence n − + n + = dim T (T M) = 2n. Let Θ be the one-form on T M determined by Θ(U ) = g(π * U, z) for all U ∈ T z M and E the "energy function" on T (T M), E(z) = 1 2 g(z, z). Definition 2.1. Adapted product structure. A product structure P in O ⊂ T M is adapted iff. PΘ = dE. (2.1) Let Σ be the geodesic spray and Ξ the Liouville (radial) vector field on T M. Recall (see [3],
more » ... Recall (see [3], [10]) the definition of horizontal and vertical lifts of a vector is the connection map. With this notation the geodesic spray Σ and the Liouville vector field Ξ are Σ(z) = (z) h z , Ξ(z) = (z) v z . (2.3) Definition 2.2. Serrate set. We call a set O ⊂ T (M) serrate if it is invariant by the contracting diffeomorphisms generated by −Σ − Ξ and Σ − Ξ. In other words, O is characterized by the following property: if γ is any unitspeed geodesic such that yγ(x) ∈ O for some x ∈ R and some y y]} ⊂ O. (See Remark 2.1.) Theorem 1. Uniqueness. In a serrate open O ⊂ T M containing M there is at most one adapted product structure P. The proof of Theorem 1 is derived from the next several propositions and follows Corollary 2.4.1, with the proof that such P always exits right afterwards. Proposition 2.1. An adapted product structure P defined on an open O ⊂ T (T M) satisfies PΣ = Ξ on O. Proof. From PdE = Θ we have d (PdE) = dΘ, and, since P is integrable, in local product coordinates there are functions t ik so that dΘ =
doi:10.4310/cag.2005.v13.n2.a6 fatcat:aksrizs3gjgnjia3o5zqoavira