## Topological Methods in Nonlinear Analysis

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

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 21, Number 2 (2003), 273-291.

Dates
First available in Project Euclid: 30 September 2016

https://projecteuclid.org/euclid.tmna/1475266299

Mathematical Reviews number (MathSciNet)
MR1998430

Zentralblatt MATH identifier
1041.53025

#### Citation

Giambò, Roberto; Giannoni, Fabio; Piccione, Paolo; Tausk, Daniel V. Morse theory for normal geodesics in sub-Riemannian manifolds with codimension one distributions. Topol. Methods Nonlinear Anal. 21 (2003), no. 2, 273--291. https://projecteuclid.org/euclid.tmna/1475266299

#### References

• A. A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity , Ann. Inst. H. Poincaré, 13 (1996), 635–690 \ref\key 2
• H. Brezis, Analyse Fonctionelle, Masson, Paris (1983) \ref\key 3
• R. Giambó, F. Giannoni and P. Piccione, Existence, multiplicity and regularity for sub-Riemannian geodesics by variational methods , SIAM J. Control Optim., 40(2002), 1840–1857 \ref\key 4
• F. Giannoni, A. Masiello, P. Piccione and D. Tausk, A Generalized Index Theorem for Morse-Sturm systems and applications to semi-Riemannian geometry , Asian J. Math., 5 (2001), 441–472 \ref\key 5
• F. Giannoni and P. Piccione, An existence theory for relativistic brachistochrones in stationary spacetimes , J. Math. Phys., 39 (1998), 6137–6152 \ref\key 6
• F. Giannoni, P. Piccione and J. A. Verderesi, An approach to the relativistic brachistochrone problem by sub–Riemannian geometry , J. Math. Phys., 38 (1997), 6367–6381 \ref\key 7
• U. Hamenstädt, Some regularity theorems for Carnot–Carathéodory metrics , J. Differential Geom., 32(1990), 819–850 \ref\key 8
• I. Kishimoto, The Morse Index Theorem for Carnot–Carathéodory spaces , J. Math. Kyoto Univ., 38 (1998), 287–293 \ref\key 9
• S. Lang, Differential Manifolds, Springer–Verlag, Berlin(1985) \ref\key 10
• W. Liu and H. J. Sussmann, Shortest paths for sub-Riemannian metrics on rank–$2$ distribution , Mem. Amer. Math. Soc., 564 (1995) \ref\key 11
• J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer–Verlag, Berlin (1989) \ref\key 12
• R. Montgomery, Singular extremals on Lie groups , Math. Control Signals, 7(1994), 217–234 \ref\key 13 ––––, Abnormal minimizers , SIAM J. Control Optim., 32 (1994), 1605–1620 \ref\key 14 ––––, A survey of singular curves in sub-Riemannian geometry , J. Dynam. Control Systems, 1 (1995), 49–90 \ref\key 15
• J. Milnor, Morse Theory, Princeton Univ. Press, Princeton (1969) \ref\key 16
• P. Piccione and D. V. Tausk, A note on the Morse Index Theorem for geodesics between submanifolds in semi-Riemannian geometry , J. Math. Phys., 40 (1999), 6682–6688 \ref\key 17 ––––, An Index Theorem for non periodic solutions of Hamiltonian systems , Proc. London Math. Soc., 83 (2001), 351–389 \ref\key 18 ––––, The Maslov index and a generalized Morse Index Theorem for non positive definite metrics , Comptes Rendus de l'Académie de Sciences de Paris, 331 (2000), 385–389 \ref\key 19 ––––, Variational aspects of the geodesic problem is sub-Riemannian geometry , J. Geom. Phys., 39(2001), 183–206 \ref\key 20 ––––, Constrained Lagragians and degenerate Hamiltonians on manifolds: an Index Theorem , Differential Equations and Dynamical Systems (Lisbon, 2000), 309–324 ;, Fields Inst. Commun., 31 (2002) \ref\key 21 ––––, On the Maslov and the Morse index for constrained variational problems , J. Math. Pures Appl. (9), 81 (2002), 403–437