Do you want to publish a course? Click here

Distribution in the unit tangent bundle of the geodesics of given type

182   0   0.0 ( 0 )
 Added by Viveka Erlandsson
 Publication date 2021
  fields
and research's language is English




Ask ChatGPT about the research

Recall that two geodesics in a negatively curved surface $S$ are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure $mathfrak{m}^S$ on $T^1S$. We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.



rate research

Read More

73 - Alberto Verjovsky 2017
Inspired by the work of Zagier, we study geometrically the probability measures $m_y$ with support on the closed horocycles of the unit tangent bundle $M=text{PSL}(2,mathbb{R})/text{PSL}(2,mathbb{Z})$ of the modular orbifold $text{PSL}(2,mathbb Z)$. In fact, the canonical projection $mathfrak{p}:Mtomathbb{H}/text{PSL}(2,mathbb Z)$ it is actually a Seifert fibration over the orbifold with two especial circle fibers corresponding to the two conical points of the modular orbifold. Zagier proved that $m_y$ converges to normalized Haar measure $m_o$ of $M$ as $yto0$: for every smooth function $f:Mto mathbb R$ with compact support $m_y(f)=m_0(f)+o(y^frac12)$ as $yto0$. He also shows that $m_y(f)=m_0(f)+o(y^{frac34-epsilon})$ for all $epsilon>0$ and smooth function $f$ with compact support in $M$ if and only if the Riemann hypothesis is true. In this paper we show that the exponent $frac12$ is optimal if $f$ is the characteristic function of certain open sets in $M$. This of course does not imply that the Riemann hypothesis is false. It is required the differentiability of the functions in the theorem.
84 - Nikolai Mnev 2006
Pure combinatorial models for BPL_n and Gauss map of a combinatorial manifold are described.
We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichm{u}ller geodesic flow.
We explicitly compute the limiting slope gap distribution for saddle connections on any 2n-gon. Our calculations show that the slope gap distribution for a translation surface is not always unimodal, answering a question of Athreya. We also give linear lower and upper bounds for number of non-differentiability points as n grows. The latter result exhibits the first example of a non-trivial bound on an infinite family of translation surfaces and answers a question by Kumanduri-Wang.
167 - Rui Albuquerque 2016
We find a new class of invariant metrics existing on the tangent bundle of any given almost-Hermitian manifold. We focus here on the case of Riemannian surfaces, which yield new examples of Kahlerian Ricci-flat manifolds in four real dimensions.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا