Morse theory for normal geodesics in sub-Riemannian manifolds with codimension one distributions

We consider a Riemannian manifold $(\mathcal M,g)$and a codimension one distribution $\Delta\subset T\mathcal M$on $\mathcal M$ which is the orthogonal of a unit vector field $Y$ on $\mathcal M$.We do not make any nonintegrability assumption on $\Delta$.The aim of the paper is to develop a Morse Theory for the sub-Riemannianaction functional $E$ on the space of horizontal curves, i.e.everywhere tangent to the distribution$\Delta$. We consider thecase of horizontal curves joining a smooth submanifold $\mathcal P$ of $\mathcal M$and a fixed point $q\in\mathcal M$. Under theassumption that $\mathcal P$ is transversal to $\Delta$, it is known (see [P. Piccione and D. V. Tausk, Variational aspects of the geodesic problem is sub-Riemannian geometry , J. Geom. Phys. 39 (2001), 183–206])that the set of such curves has the structure of an infinite dimensionalHilbert manifold and that the critical points of $E$ are the so called{\it normal extremals} (see [W. Liu and H. J. Sussmann, Shortest paths for sub-Riemannian metrics on rank–$2$distribution , Mem. Amer. Math. Soc. 564 (1995)]). We compute thesecond variation of $E$ at its critical points, we definethe notions of $\mathcal P$-Jacobi field, of $\mathcal P$-focal pointand of exponential map and we prove a MorseIndex Theorem. Finally, we prove the Morse relations for the critical points of $E$ under the assumptionof completeness for $(\mathcal M,g)$.