Morse Theory for Normal Geodesics in Sub-riemannian Manifolds with Codimension One Distributions

We consider a Riemannian manifold (M, g) and a codimension one distribution ∆ ⊂ TM on M which is the orthogonal of a unit vector field Y onM. We do not make any nonintegrability assumption on ∆. The aim of the paper is to develop a Morse Theory for the sub-Riemannian action functional E on the space of horizontal curves, i.e. everywhere tangent to the distribution ∆. We consider the case of horizontal curves joining a smooth submanifold P of M and a fixed point q ∈ M. Under the assumption that P is transversal to ∆, it is known (see [19]) that the set of such curves has the structure of an infinite dimensional Hilbert manifold and that the critical points of E are the so called normal extremals (see [10]). We compute the second variation of E at its critical points, we define the notions of P-Jacobi field, of P-focal point and of exponential map and we prove a Morse Index Theorem. Finally, we prove the Morse relations for the critical points of E under the assumption of completeness for (M, g).