Showing papers on "Unit tangent bundle published in 2019"
TL;DR: It is presented strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesICS.
Abstract: Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.
10 citations
TL;DR: In this paper, contact magnetic curves on the unit tangent bundle UM of a Riemannian manifold were investigated and the magnetic equations were written in terms of contact normal magnetic curves.
Abstract: In this paper we investigate contact magnetic curves on the unit tangent bundle UM of a Riemannian manifold M and we write the magnetic equations. In the case when M is a space form M(c), it follows that every contact normal magnetic curve is slant when $$c=1$$
, while for $$c
e 1$$
a contact normal magnetic curve is slant if and only if it satisfies a conservation law. We perform a detailed study of contact magnetic curves in $$U{{\mathbb {S}}}^2$$
.
10 citations
TL;DR: In this paper, an outer space analogue for the principal stratum of the unit tangent bundle to the Teichmuller space was proposed, where the geodesics in Teichmeiler space were determined by the principal stratatum.
Abstract: In this paper we propose an Outer space analogue for the principal stratum of the unit tangent bundle to the Teichmuller space $\mathcal{T}(S)$ of a closed hyperbolic surface $S$. More specifically, we focus on properties of the geodesics in Teichmuller space determined by the principal stratum. We show that the analogous Outer space "principal" periodic geodesics share certain stability properties with the principal stratum geodesics of Teichmuller space. We also show that the stratification of periodic geodesics in Outer space exhibits some new pathological phenomena not present in the Teichmuller space context.
9 citations
TL;DR: In this article, the authors give an upper bound for the volume of a periodic geodesic on a surface with respect to the unit tangent bundle, which is linear in the geometric length.
Abstract: A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer. We give an upper bound for this volume which is linear in the geometric length of the geodesic.
8 citations
TL;DR: For families of hypersurfaces with singular points, a classical definition of an envelope is vague as mentioned in this paper ; however, it is known that the projection of a singular solution of a first order partial differential equation is an envelope.
8 citations
TL;DR: In this paper, the authors introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian two-manifold, which can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties.
Abstract: We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian two-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We show that the family of flows can be parametrised in terms of certain weighted holomorphic differentials and investigate their properties. In particular, we prove that they admit a dominated splitting and we identify special cases in which the flows are Anosov. In the latter case, we study when they admit an invariant measure in the Lebesgue class and the regularity of the weak foliations.
7 citations
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TL;DR: In this article, it was shown that for closed surfaces with Riemannian metrics without conjugate points, the geodesic flow on the unit tangent bundle has a unique measure of maximal entropy.
Abstract: We prove that for closed surfaces $M$ with Riemannian metrics without conjugate points and genus $\geq 2$ the geodesic flow on the unit tangent bundle $T^1M$ has a unique measure of maximal entropy. Furthermore, this measure is fully supported on $T^1M$ and the flow is mixing with respect to this measure. We formulate conditions under which this result extends to higher dimensions.
6 citations
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TL;DR: In this paper, the authors characterize the corresponding divide monodromies as exactly the products of a left-veering horizontal and a rightveering vertical antitwist with respect to a cylinder decomposition, where the notion of an antitwists is an orientation-reversing analogue of a multitwist.
Abstract: A divide on an orientable 2-orbifold gives rise to a fibration of the unit tangent bundle to the orbifold.We characterize the corresponding monodromies as exactly the products of a left-veering horizontal and a right-veering vertical antitwist with respect to a cylinder decomposition, where the notion of an antitwist is an orientation-reversing analogue of a multitwist. Many divide monodromies are pseudo-Anosov and we give plenty of this http URL particular, we show that there exist divide monodromies with stretch factor arbitrarily close to one, and give an example none of whose powers can be obtained by Penner's or Thurston's construction of pseudo-Anosov mapping this http URL a side product, we also get a new combinatorial construction of pseudo-Anosov mapping classes in terms of products of antitwists.
3 citations
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TL;DR: In this article, the brain tends to complete the missing contour of a picture using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1.
Abstract: If a piece of the contour of a picture is missing to the eye vision, then the brain tends to complete it using some kind of sub-Riemannian geodesics of the unit tangent bundle of the plane, R2xS1 These geodesics can be obtained by lifting extremal curves of a total curvature type energy in the plane We completely solve this variational problem, geometrically Moreover, we also show a way of constructing rotational surfaces of constant negative curvature in R3 by evolving these extremal curves under their associated binormal flow with prescribed velocity Finally, we prove that, locally, all rotational constant negative curvature surfaces of R3 are foliated by extremal curves of these energies Therefore, we conclude that there exists a one-to-one correspondence between the sub-Riemannian geodesics used by the brain for visual curve completion and these rotational surfaces of R3
2 citations
TL;DR: In this paper, effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces was proved.
Abstract: We prove effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces.
2 citations
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TL;DR: In this article, it was shown that every geodesic flow on the unit tangent bundle of a negatively curved 2-dimensional orbifold is almost equivalent to the suspension of some automorphism of the torus.
Abstract: Two flows on two compact manifolds are almost equivalent if there is a homeomorphism from the complement of a finite number of periodic orbits of the first flow to the complement of the same number of periodic orbits of the second flow that sends orbits onto orbits. We prove that every geodesic flow on the unit tangent bundle of a negatively curved 2-dimensional orbifold is almost equivalent to the suspension of some automorphism of the torus. Together with a result of Minikawa, this implies that all algebraic Anosov flows are pairwise almost equivalent. We initiate the study of the Ghys graph ---an analogue of the Gordian graph in this context-by giving explicit upper bounds on the distances between these flows.
TL;DR: In this article, the authors investigated the stability and instability of harmonic unit vector fields on compact quotients of four dimensional oscillator group G1(1) and determined the associated harmonic maps from the oscillator groups into its unit tangent bundle equipped with the associated Sasaki metric.
Abstract: We find all the left-invariant harmonic unit vector fields on the oscillator groups. Besides, we determine the associated harmonic maps from the oscillator group into its unit tangent bundle equipped with the associated Sasaki metric. Moreover, we investigate the stability and instability of harmonic unit vector fields on compact quotients of four dimensional oscillator group G1(1).
27 Mar 2019
TL;DR: In this article, a one-to-one correspondence between the set of unit split semi-quaternions and unit tangent bundle of semi-Euclidean plane is given.
Abstract: In this paper, a one-to-one correspondence between the set of unit split semi-quaternions and unit tangent bundle of semi-Euclidean plane is given It is shown that the set of unit split semiquaternions based on the group operation of multiplication is a Lie group The Lie algebra of this group, consisting of the vector space matrix of the angular velocity vectors, is also considered Planar rotations in Euclidean plane are expressed using split semi-quaternions Some examples are given to illustrate the findings
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TL;DR: In this paper, effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces was proved.
Abstract: We prove effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces.
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TL;DR: In this article, a complete description of the hyperplane section of the projective bundle associated to the tangent bundle under its natural embedding in $\mathbb{P}^7.$ is given.
Abstract: In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$