Related papers: Integrable sub-Riemannian geodesic flows on the sp…

We prove that if the normal distribution of a singular riemannian foliation is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section) which meets every…

We introduce a special class of nilpotent Lie groups of step 2, that generalizes the so called $H$(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of inner product to an arbitrary scalar product and relate the…

This paper proposes an original Riemmanian geometry for low-rank structured elliptical models, i.e., when samples are elliptically distributed with a covariance matrix that has a low-rank plus identity structure. The considered geometry is…

This paper is a review of recent and classical results on integrable geodesic flows on Riemannian manifolds and topological obstructions to integrability. We also discuss some open problems.

The geodesic orbit property has been studied intensively for Riemannian manifolds. Geodesic orbit spaces are homogeneous and allow simplifications of many structural questions using the Lie algebra of the isometry group. Weakly symmetric…

In this paper we construct smooth Riemannian metrics on the sphere which admit smooth Zoll families of minimal hypersurfaces. This generalizes a theorem of Guillemin for the case of geodesics. The proof uses the Nash-Moser Inverse Function…

In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics,…

We study isometric immersions of surfaces into simply connected 3-dimensional unimodular Lie groups endowed with either Riemannian or Lorentzian left-invariant metrics, assuming that Milnor's operator is diagonalizable in the Lorentzian…

In the sub-Riemannian Heisenberg group equipped with its Carnot-Caratheodory metric and with a Haar measure, we consider isodiametric sets, i.e. sets maximizing the measure among all sets with a given diameter. In particular, given an…

We study the existence of closed geodesics on compact Riemannian orbifolds, and on noncompact Riemannian manifolds in the presence of a cocompact, isometric group action. We show that every noncontractible Riemannian manifold which admits…

We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case $\Emb(\Bbb R,\Bbb R)$ which turns out to be Burgers'…

We construct convex bodies that can be "captured by nets." More precisely, for each dimension $n \geq 2$, we construct a family of Riemannian $n$-spheres, each with a stable geodesic net, which is a stable 1-dimensional integral varifold.…

The paper is devoted to the study of geodesic orbit Riemannian metrics on nilpotent Lie groups. The main result is the construction of continuous families of pairwise non-isomorphic connected and simply connected nilpotent Lie groups, every…

We construct harmonic Riemannian submersions that are retractions from symmetric spaces of noncompact type onto their rank-one totally geodesic subspaces. Among the consequences, we prove the existence of a non-constant, globally defined…

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…

In this paper, we discuss the decomposition of a Lie group with a left invariant pseudo-Riemannian metric and the uniqueness. In fact, it is a decomposition of a Lie group into totally geodesic sub-manifolds which is different from the De…

We prove that some symetric semi-riemannian manifolds do not admit a proper domain which is divisible by the action of a discrete group of isometries. In other words, if a closed semi-riemannian manifold is locally isometric to such a…

To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on…

In this note we give a construction of a smooth Riemannian metric on R^n which is standard Euclidean outside a compact set K and such that it has N = n(n + 1)=2 invisible directions, meaning that all geodesics lines passing through the set…

We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space…