On the measures of maximal entropy for the geodesic flow

Let M be a closed and connected manifold equipped with a C^∞ Riemannian metric. Using the geodesic arc formalism developed by Maé (J. Differential Geom. 45 (1997) 74–93) and Paternain [Ann. Sci. École Norm. Sup. 4eme série t 33 (2000) 121–138], we give a way of constructing a measure of maximal entropy for the geodesic flow.     

Obstructions to toric integrable geodesic flows in dimension 3

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T ^* Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10     

Scattering metrics and geodesic flow at infinity

Any compact ? ^∞ manifold with boundary admits a Riemann metric on its interior taking the form x ⁻⁴ dx ² +x ⁻² h′ near the boundary, where x is a boundary defining function and h′ is a smooth symmetric 2-cotensor restricting to be positive-definite, and hence a metric, h, on the boundary. The scattering theory associated to the Laplacian for such a ‘scattering metric’ was discussed by the first author and here it is shown, as conjectured, that the scattering matrix is a Fourier integral operator which quantizes the geodesic flow on the boundary, for the metric h, at time π. To prove this the Poisson operator, of the associated generalized boundary problem, is constructed as a Fourier integral operator associated to a singular Legendre manifold. Oblatum 24-VII-1995     

Totally geodesic maps into metric spaces

We prove that a totally geodesic map between a Riemannian manifold and a metric space can be represented as the composite of a totally geodesic map from a Riemannian manifold to a Finslerian manifold and a locally isometric embedding between metric spaces. As a corollary, we obtain the homotheticity of a totally geodesic map from an irreducible Riemannian manifold to an Alexandrov space of curvature bounded above. This is a generalization of the case between Riemannian manifolds. Mathematics Subject Classification (2000): 53C20, 53C22, 53C24 Received: 14 March 2002; in final form: 6 May 2002 / / Published online: 24 February 2003     

Regularity of the Anosov distributions of Euler-Lagrange deformations of geodesic flows

Let g be a negatively curved Riemannian metric of a closed C ^∞ manifold M of dimension at least three. Let L _λ be a C ^∞ one-parameter convex superlinear Lagrangian on TM such that L₀(v) = \frac12 g(v, v){L_0(v)= \frac{1}{2} g(v, v)} for any v ∈ TM. We denote by j^l{\varphi^\lambda} the restriction of the Euler-Lagrange flow of L _λ on the \frac12{\frac{1}{2}} -energy level. If λ is small enough then the flow j^l{\varphi^\lambda} is Anosov. In this paper we study the geometric consequences of different assumptions about the regularity of the Anosov distributions of j^l{\varphi^\lambda} . For example, in the case that the initial Riemannian metric g is real hyperbolic, we prove that for λ small, j^l{\varphi^\lambda} has C ³ weak stable and weak unstable distributions if and only if j^l{\varphi^\lambda} is C ^∞ orbit equivalent to the geodesic flow of g.     

Orbits of the geodesic flow and chains on the boundary of a Grauert tube

A Riemannian manifold (X,g) determines an integrable complex structure on a tubular neighborhood, , of the zero section in and a CR-structure on the boundary, . There are two natural families of curves on : the orbits of the geodesic flow and a CR-invariant family called chains. It is natural to ask whether they are related. We show that if orbits of the geodesic flow are chains on for all sufficiently small, then (X,g) is Einstein. As a partial converse we show that if (X,g) is harmonic, then orbits of the geodesic flow are chains. To prove this we study the Fefferman metric associated with . Received: 1 April 1998 / Revised version: 25 May 2001 / Published online: 19 October 2001     

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

Given a rational homology classh in a two dimensional torusT ², we show that the set of Riemannian metrics inT ² with no geodesic foliations having rotation numberh isC ^k dense for everyk N. We also show that, generically in theC ² topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.Partially supported by CNPq-GMD, FAPERJ, and the University of Freiburg.     

Orbits of unbounded energy in quasi-periodic perturbations of geodesic flows

We show that certain mechanical systems, including a geodesic flow in any dimension plus a quasi-periodic perturbation by a potential, have orbits of unbounded energy.The assumptions we make in the case of geodesic flows are:

(a)

The metric and the external perturbation are smooth enough.

(b)

The geodesic flow has a hyperbolic periodic orbit such that its stable and unstable manifolds have a tranverse homoclinic intersection.

(c)

The frequency of the external perturbation is Diophantine.

(d)

The external potential satisfies a generic condition depending on the periodic orbit considered in (b).

The assumptions on the metric are C² open and are known to be dense on many manifolds. The assumptions on the potential fail only in infinite codimension spaces of potentials.The proof is based on geometric considerations of invariant manifolds and their intersections. The main tools include the scattering map of normally hyperbolic invariant manifolds, as well as standard perturbation theories (averaging, KAM and Melnikov techniques).We do not need to assume that the metric is Riemannian and we obtain results for Finsler or Lorentz metrics. Indeed, there is a formulation for Hamiltonian systems satisfying scaling hypotheses. We do not need to make assumptions on the global topology of the manifold nor on its dimension.     

Harmonicity of totally geodesic maps into nonpositively curved metric spaces

We prove the harmonicity of totally geodesic maps from a Riemannian manifold to a nonpositively curved metric space in the sense of Alexandrov for both Korevaar-Schoen-type and Cheeger-type energies. This enables us to make many examples of harmonic maps of an unknown type. We also construct an example of totally geodesic map between CAT(0)-spaces which is not harmonic.Mathematics Subject Classification (2000): 53C22, 53C43, 58E20     

Self-intersecting geodesics and entropy of the geodesic flow

Given a symmetric Finsler metric on T^2 whose geodesic flow has zero topological entropy, we show that the lift in the universal covering R^2 →T^2 of any closed geodesic on T^2 must be an embedded curve in R^2.     

Totally geodesic submanifolds in Lie groups

Summary We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G_F(n,N) in the Lie group G_F(N) defined naturally makes G_F(n,N) a totally geodesic submanifold; (2) The imbedding S⁷→SO(8) defined by octonians makes S⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G_F(N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G_F(N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F)is formed by the eigenfunctions of the Laplacian on G_F(N).     

Invariant fibrations of geodesic flows

Let (Σ,g) be a compact C² finslerian 3-manifold. If the geodesic flow of g is completely integrable, and the singular set is a tamely-embedded polyhedron, then π₁(Σ) is almost polycyclic. On the other hand, if Σ is a compact, irreducible 3-manifold and π₁(Σ) is infinite polycyclic while π₂(Σ) is trivial, then Σ admits an analytic riemannian metric whose geodesic flow is completely integrable and singular set is a real-analytic variety. Additional results in higher dimensions are proven.     

Normal and tangential geodesic deformations of the surfaces of revolution

We study special infinitesimal geodesic deformations of the surfaces of revolution in the Euclidean space E ³.     

The characterization of totally geodesic submanifolds ofS m andCP m

We prove a theorem on ruled surfaces that generalizes a theorem of Ferus on totally geodesic foliations. On the basis of this theorem we obtain criteria for totally geodesic submanifolds ofS ^m andCP ^m that generalize and complement certain results of Borisenko, Ferus, and Abe. We give an application to the geodesic differential forms defined by Dombrowski in the case of submanifolds ofS ^m andCP ^m.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 106–116.The author is grateful to V. A. Toponogov for posing this problem and for attention to the work and to A. A. Borisenko for helpful criticisms.     

Derivatives with respect to metrics and applications: subgradient marching algorithm

This paper introduces a subgradient descent algorithm to compute a Riemannian metric that minimizes an energy involving geodesic distances. The heart of the method is the Subgradient Marching Algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in O(N ² log(N)) operations on a discrete grid of N points. It performs a front propagation that computes a subgradient of a discrete geodesic distance. We show applications to landscape modeling and to traffic congestion. Both applications require the maximization of geodesic distances under convex constraints, and are solved by subgradient descent computed with our Subgradient Marching. We also show application to the inversion of travel time tomography, where the recovered metric is the local minimum of a non-convex variational problem involving geodesic distances.     

Partially hyperbolic geodesic flows

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.     

The furthest-site geodesic voronoi diagram

We present anO((n+k) log(n+k))-time,O(n+k)-space algorithm for computing the furthest-site Voronoi diagram ofk point sites with respect to the geodesic metric within a simplen-sided polygon.A preliminary version of this paper appeared in theProceedings of the Fourth ACM Symposium on Computational Geometry, 1988. The work of Boris Aronov was supported by an AT&T Bell Laboratories Ph.D. Scholarship. Part of the work was performed while he was at AT&T Bell Laboratories, Murray Hill, NJ, USA. His current address is Computer Science Department, Polytechnic University, Brooklyn, NY 11201, USA.     

On totally geodesic foliations with bundle-like metric

Let be a totally geodesic foliation of dimension n and codimension p on a Riemannian manifold (M, g). Suppose that g is a bundle-like metric for and M has at least one point at which none of its mixed sectional curvatures vanishes. Under these conditions we prove that n ≤ p − 1. We show that this inequality is optimal, and none of the above conditions can be removed.     

On geodesic exponential maps of the Virasoro group

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ^(k) (k≥ 0) on the Virasoro group Vir and show that for k≥ 2, but not for k = 0,1, each of them defines a smooth Fréchet chart of the unital element e ∈Vir. In particular, the geodesic exponential map corresponding to the Korteweg–de Vries (KdV) equation (k = 0) is not a local diffeomorphism near the origin. A. Constantin: Supported in part by the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652). T. Kappeler: Supported in part by the SNSF, the programme SPECT, and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652)     

Liouville and geodesic Ricci solitons

On a tangent bundle endowed with a pseudo-Riemannian metric of complete lift type two classes of Ricci solitons are obtained: a 1-parameter family of shrinking Liouville Ricci solitons if the base manifold is Ricci flat and a steady geodesic Ricci soliton if the base manifold is flat. A nonexistence result of geodesic Ricci solitons for the tangent bundle of a non-flat space form is also provided. To cite this article: M. Crasmareanu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).