Morse theory for normal geodesics in sub-Riemannian manifolds with codimension one distributions (original) (raw)

Genericity of nondegenerate critical points and Morse geodesic functionals

Indiana University Mathematics Journal, 2009

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard-Smale theorem, along the lines of an analogous result of B. White [27]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.

End-Point Equations and Regularity of Sub-Riemannian Geodesics

Geometric and Functional Analysis, 2008

For a large class of equiregular sub-Riemannian manifolds, we show that length-minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank-2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x 1 , x 2 . Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples.

Morse-Sard type results in sub-Riemannian geometry

Mathematische Annalen, 2005

Let (M, ∆, g) be a sub-Riemannian manifold and x0 ∈ M . Assuming that Chow's condition holds and that M endowed with the sub-Riemannian distance is complete, we prove that there exists a dense subset N1 of M such that for every point x of N1, there is a unique minimizing path steering x0 to x, this trajectory admitting a normal extremal lift. If the distribution ∆ is everywhere of corank one, we prove the existence of a subset N2 of M of full Lebesgue measure such that for every point x of N2, there exists a minimizing path steering x0 to x which admits a normal extremal lift, is nonsingular, and the point x is not conjugate to x0. In particular, the image of the sub-Riemannian exponential mapping is dense in M , and in the case of corank one is of full Lebesgue measure in M .

A generalized index theorem for Morse-Sturm systems and applications to semi-Riemannian geometry

2001

We prove an extension of the Index Theorem for Morse-Sturm systems of the form −V ′′ + RV = 0, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M, g), we consider a geodesic γ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M, g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at γ and the sum of the Maslov index of γ with the index of the metric g on P. Under generic circumstances, the Maslov index of γ is given by an algebraic count of the P-focal points along γ. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold.

Critical Point Theory and Submanifold Geometry

1988

This book is divided into two parts. Part I is a modern introduction to the very classical theory of submanifold geometry. We go beyond the classical theory in at least one important respect; we study submanifolds of Hilbert space as well as of Euclidean spaces. Part II is devoted to critical point theory, and here again the theory is developed in the setting of Hilbert manifolds. The two parts are inter-related through the Morse Index Theorem, that is, the fact that the structure of the set of critical points of the distance function from a point to a submanifold can be described completely in terms of the local geometric invariants of the submanifold.

A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds

Topological Methods in Nonlinear Analysis, 2005

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse index by its relative form, given by the spectral flow of an associated family of index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context, and prove that our generalized Morse index equals the total number of conjugate points. Finally we study the relation with the Maslov index of the flow induced on the Lagrangian Grassmannian.

A Morse Index Theorem and bifurcation for perturbed geodesics on Semi-Riemannian Manifolds

2003

Perturbed geodesics are trajectories of particles moving on a semi-Riemannian manifold in the presence of a potential. Our purpose here is to extend to perturbed geodesics on semi-Riemannian manifolds the well known Morse Index Theorem. When the metric is indefinite, the Morse index of the energy functional becomes infinite and hence, in order to obtain a meaningful statement, we substitute the Morse index by its relative form, given by the spectral flow of an associated family of index forms. We also introduce a new counting for conjugate points, which need not to be isolated in this context, and prove that the relative Morse index equals the total number of conjugate points. Finally we study the relation with the Maslov index of the flow induced on the Lagrangian Grassmannian.

Characterizations of hamiltonian geodesics in sub-riemannian geometry

Journal of Dynamical and Control Systems, 1997

Let (M, E, g) be a sub-Riemannian manifold. The geodesics of (M, E, g) are either Hamiltonian or srictly abnormal. In this paper we give necessary and sufficient conditions for a geodesic curve to be Hamiltonian, and we apply this to the study of totally geodesic submanifolds of Lie groups. We also give an example of a foliation by totally geodesic submanifolds, each leaf containing a one parameter family of geodesics which are Hamiltonian in the leaf but stricly abnormal in the ambient space.

Morse index and bifurcation of p-geodesics on semi Riemannian manifolds

Esaim-control Optimisation and Calculus of Variations, 2007

Given a one-parameter family {g λ : λ ∈ [a, b]} of semi Riemannian metrics on an ndimensional manifold M , a family of time-dependent potentials {V λ : λ ∈ [a, b]} and a family {σ λ : λ ∈ [a, b]} of trajectories connecting two points of the mechanical system defined by (g λ , V λ ), we show that there are trajectories bifurcating from the trivial branch σ λ if the generalized Morse indices µ(σa) and µ(σ b ) are different.