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

DOI: http://dx.doi.org/10.12775/TMNA.2003.016

#### 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 $(\mathcal M,g)$.

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 $(\mathcal M,g)$.

#### Keywords

Morse theory; sub-Riemannian geometry; nomal sub-Remannian minimizers

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