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

Abstract

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-Riemannian action functional $E$ on the space of horizontal curves, i.e. everywhere tangent to the distribution $\Delta$. We consider the case of horizontal curves joining a smooth submanifold $\mathcal P$ of $\mathcal M$ and a fixed point $q\in\mathcal M$. Under the assumption 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 dimensional Hilbert manifold and that the critical points of $E$ are the so called 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 the second variation of $E$ at its critical points, we define the notions of $\mathcal P$-Jacobi field, of $\mathcal P$-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 $(\mathcal M,g)$.