Scattering matrix for asymptotically euclidean manifolds
Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups.
Shells of monotone curves
We determine in the form of curves corresponding to strictly monotone functions as well as the components of affine connections for which any image of under a compact-free group of affinities containing the translation group is a geodesic with respect to . Special attention is paid to the case that contains many dilatations or that is a curve in . If is a curve in and is the translation group then we calculate not only the components of the curvature and the Weyl tensor but...
Simplicial nonpositive curvature
We introduce a family of conditions on a simplicial complex that we call local k-largeness (k≥6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher...
Small eigenvalues of Riemann surfaces and graphs.
Smooth quasigroups and loops: forty-five years of incredible growth
The remarkable development of the theory of smooth quasigroups is surveyed.
Some existence theorems for closed geodesics.
Some isoperimetric inequalities and eigenvalue estimates
Some properties of geodesic semi E-b-vex functions
In this study, we introduce a new class of function called geodesic semi E-b-vex functions and generalized geodesic semi E-b-vex functions and discuss some of their properties.
Spectrum of a compact flat manifold.
Spectrum of the Laplace operator and periodic geodesics: thirty years after
What is called the “Semi-classical trace formula” is a formula expressing the smoothed density of states of the Laplace operator on a compact Riemannian manifold in terms of the periodic geodesics. Mathematical derivation of such formulas were provided in the seventies by several authors. The main goal of this paper is to state the formula and to give a self-contained proof independent of the difficult use of the global calculus of Fourier Integral Operators. This proof is close in the spirit of...
Stability of the geodesic flow for the energy
We study the stability of the geodesic flow as a critical point for the energy functional when the base space is a compact orientable quotient of a two-point homogeneous space.
Strongly geodesically automatic groups are hyperbolic.
Structure equations on generalized Finsler manifolds
In this paper we generalize the classical structure equations of Riemannian geometry to generalized Finsler manifolds.
Structure of geodesics in weakly symmetric Finsler metrics on H-type groups
Structure of geodesic graphs in special families of invariant weakly symmetric Finsler metrics on modified H-type groups is investigated. Geodesic graphs on modified H-type groups with the center of dimension or are constructed. The new patterns of algebraic complexity of geodesic graphs are observed.
Subriemannian geodesics of Carnot groups of step 3
In Carnot groups of step ≤ 3, all subriemannian geodesics are proved to be normal. The proof is based on a reduction argument and the Goh condition for minimality of singular curves. The Goh condition is deduced from a reformulation and a calculus of the end-point mapping which boils down to the graded structures of Carnot groups.
Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity
Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity
We study sub-Riemannian (Carnot-Caratheodory) metrics defined by noninvolutive distributions on real-analytic Riemannian manifolds. We establish a connection between regularity properties of these metrics and the lack of length minimizing abnormal geodesics. Utilizing the results of the previous study of abnormal length minimizers accomplished by the authors in [Annales IHP. Analyse nonlinéaire 13, p. 635-690] we describe in this paper two classes of the germs of distributions (called 2-generating...
Sub-riemannian sphere in Martinet flat case
Sub-Riemannian sphere in Martinet flat case
This article deals with the local sub-Riemannian geometry on ℜ3, (D,g) where D is the distribution ker ω, ω being the Martinet one-form : dz - ½y2dxand g is a Riemannian metric on D. We prove that we can take g as a sum of squares adx2 + cd2. Then we analyze the flat case where a = c = 1. We parametrize the set of geodesics using elliptic integrals. This allows to compute the exponential mapping, the wave front, the conjugate and cut loci and the sub-Riemannian sphere. A direct consequence...