For any ε > 0, we construct an explicit smooth Riemannian metric on the sphere Sn, n ≥ 3, that is within ε of the round metric and has a geodesic for which the corresponding orbit of the geodesic flow is ε-dense in the unit tangent bundle. Moreover, for any ε > 0, we construct a smooth Riemannian metric on S, n ≥ 3, that is within ε of the round metric and has a geodesic for which the complemen...

In this paper we study a Riemanian metric on the tangent bundle T (M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T (M) a structure of locally conformal almost Kählerian manifold. This is the natural generalization of the well known almost Kählerian structure on T (M). W...

Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously some regimes. Here we provide formal results for original regime [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049-3052, 1998]. We first construct a trapping region...

Abstract We introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of singular subanalytic germs. The main novelty our approach is allow “moderately discontinuous” chains, are specially advantageous for capturing the subtleties outer phenomena. Our invariant finitely generated graded abelian group any and homomorphisms ....

In this note, we show that the holonomy group of a Riemannian connection on a k-dimensional Euclidean vector bundle is transitive on the unit sphere bundle whenever the Euler class χ is spherical. We extract several consequences from this, among them that this is always the case as long as χ does not vanish, and the base of the bundle is simply connected and rationally (k + 1)/2-connected.

Any line is associated with a two-parameter group of polar complexes relative to a given first-order line complex that are called tangential complexes when the line belongs to the complex. The congruence that is determined by the tangential complex of a complex line is of a special kind, in that both of its directrices coincide in a single line that is the complex line itself. It is of a specia...

In this study, we give the relationships between the conical curvatures of ruled surfaces generated by the unit vectors of the ruling, central normal and central tangent of a ruled surface in the Euclidean 3-space E^3. We obtain differential equations characterizing slant ruled surfaces and if the reference ruled surface is a slant ruled surface, we give the conditions for the surfaces generate...

Contents 1. Introduction 2 1.1. Equidistribution theorems for closed geodesics 2 1.2. The modular surface 3 1.3. Results 5 1.4. Remarks 7 1.5. Plan of the paper 9 1.6. Acknowledgments 9 2. Background on periods 9 2.1. The upper half-plane and its unit tangent bundle 9 2.2. Quotients 11 2.3. A correspondence with binary quadratic forms 12 2.4.