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, < i> Variational aspects of the geodesic problem is
sub-Riemannian geometry< /i> , J. Geom. Phys. < b> 39< /b> (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
{\it normal extremals} (see [W. Liu and H. J. Sussmann, < i> Shortest paths for sub-Riemannian metrics on rank–$2$
distribution< /i> , Mem. Amer. Math. Soc. < b> 564< /b> (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 $(\mathcal M,g)$.