Showing papers on "Unit tangent bundle published in 2017"

Invariant distributions and X-ray transform for Anosov flows

Abstract: For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $\Pi$ which maps into the space of flow invariant distributions in $\cap_{r \lt 0} H^r (\mathcal{M})$ and whose kernel is made of coboundaries in $\cup_{s \gt 0} H^s (\mathcal{M})$. We describe relations to the Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{M}= SM$ of a compact manifold $\mathcal{M}$, we apply this theory to study X-ray transform on symmetric tensors on $\mathcal{M}$. In particular, we prove existence of flow invariant distributions on $SM$ with prescribed push-forward on $\mathcal{M}$ and a similar version for tensors. This allows us to show injectivity of the X-ray transform on an Anosov surface: any divergence-free symmetric tensor on $\mathcal{M}$ which integrates to $0$ along all closed geodesics is zero.

Ruled Surfaces and Tangent Bundle of Unit 2-Sphere

Abstract: In this paper, a one-to-one correspondence is given between the tangent bundle of unit 2-sphere, T𝕊2, and the unit dual sphere, 𝕊𝔻2 According to Study’s map, to each curve on 𝕊𝔻2 corresponds a rul

An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications

Abstract: In a previous work, the authors introduced the notion of ‘coherent tangent bundle’, which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss–Bonnet formulas on coherent tangent bundles on $2$-dimensional manifolds were proven, and several applications to surface theory were given. Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal{E}$ be a vector bundle of rank $n$ over $M^n$, and $\phi:TM^n\to \mathcal{E}$ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss–Bonnet formulas can be generalized to an index formula for the bundle homomorphism $\phi$ under the assumption that $\phi$ admits only certain kinds of generic singularities. We shall give several applications to hypersurface theory. Moreover, as an application for intrinsic geometry, we also give a characterization of the class of positive semi-definite metrics (called Kossowski metrics) which can be realized as the induced metrics of the coherent tangent bundles.

Envelopes of Legendre Curves in the Unit Tangent Bundle over the Euclidean Plane

Abstract: For singular plane curves, the classical definitions of envelopes are vague. In order to define envelopes for singular plane curves, we introduce a one-parameter family of Legendre curves in the unit tangent bundle over the Euclidean plane and the curvature. Then we give a definition of an envelope for the one-parameter family of Legendre curves. We investigate properties of the envelopes. For instance, the envelope is also a Legendre curve. Moreover, we consider bi-Legendre curves and give a relationship between envelopes.

Equivariant maps into Anti-de Sitter space and the symplectic geometry of ℍ²×ℍ²

Abstract: Given two Fuchsian representations ρ l and ρr of the fundamental group of a closed oriented surface S of genus ≥ 2, we study the relation between Lagrangian submanifolds of Mρ = (H^2/ρ_l(π_1(S))) × (H^2/ρ_r(π_1(S))) and ρ-equivariant embeddings σ of S into Anti-de Sitter space, where ρ = (ρ_l ,ρ_r) is the corresponding representation into PSL(2,R) × PSL(2,R). It is known that, if σ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold Λ ML of Mρ. We show that, given any ρ-equivariant embedding σ, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to Λ_ML. Conversely, any Lagrangian submanifold Hamiltonian isotopic to Λ_ML is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.

New Cases of Integrable Systems with Dissipation on a Tangent Bundle of a Two-Dimensional Manifold

Abstract: The integrability of certain classes of dynamic systems is shown on the tangent bundles to twodimensional manifolds. In this case, the force fields have the so-called variable dissipation and generalize the previously considered fields.

N-Legendre and N-slant curves in the unit tangent bundle of surfaces

Abstract: Let (T1M; g1) be a unit tangent bundle of some surface (M; g) en-dowed with the induced Sasaki metric. In this present paper, we de-…ne two kinds of curves called N-legendre and N-slant curves as curveshaving an inner product of normal vector and Reeb vector zero andnonzero constant respectively and several important characterizationsof these curves are obtained.

Global linearization and fiber bundle structure of invariant manifolds

Abstract: We study global properties of the global (center-)stable manifold of a normally attracting invariant manifold (NAIM), the special case of a normally hyperbolic invariant manifold (NHIM) with empty unstable bundle. We show that the global stable foliation of a NAIM has the structure of a topological disk bundle, and that similar statements hold for inflowing NAIMs and for general compact NHIMs. Furthermore, the global stable foliation has a $C^k$ disk bundle structure if the local stable foliation is assumed $C^k$. We then show that the dynamics restricted to the stable manifold of a compact inflowing NAIM are globally topologically conjugate to the linearized transverse dynamics at the NAIM. Moreover, we give conditions ensuring the existence of a global $C^k$ linearizing conjugacy. We illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.

Holomorphic differentials, thermostats and Anosov flows

Abstract: We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian $2$-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.

Gibbs u -states for the foliated geodesic flow and transverse invariant measures

Abstract: This paper is devoted to the study of Gibbs u-states for the geodesic flow tangent to a foliation F of a manifold M having negatively curved leaves. By definition, they are the probability measures on the unit tangent bundle to the foliation that are invariant under the foliated geodesic flow and have Lebesgue disintegration in the unstable manifolds of this flow. p]On the one hand we give sufficient conditions for the existence of transverse invariant measures. In particular we prove that when the foliated geodesic flow has a Gibbs su-state, i.e. an invariant measure with Lebesgue disintegration both in the stable and unstable manifolds, then this measure has to be obtained by combining a transverse invariant measure and the Liouville measure on the leaves. p]On the other hand we exhibit a bijective correspondence between the set of Gibbs u-states and a set of probability measure on M that we call φ u -harmonic. Such measures have Lebesgue disintegration in the leaves and their local densities have a very specific form: they possess an integral representation analogue to the Poisson representation of harmonic functions.

Coordinate representation of the Lagrange-Poincar\'{e} equations for a mechanical system with symmetry on the total space of a principal fiber bundle whose base is the bundle space of the associated bundle

Abstract: Using the dependent coordinates, the local Lagrange-Poincare equations and equations for the relative equilibria are obtained for a mechanical system with a symmetry describing the motion of two interacting scalar particles on a special Riemannian manifold (the product of the total space of the principal fiber bundle and the vector space) on which a free proper and isometric action of a compact semi-simple Lie group is given As in gauge theories, dependent coordinates are implicitly determined by means of equations representing the local sections of the principal fiber bundle

Maxwell's equations in media as a contact Hamiltonian vector field and its information geometry -- An approach with a bundle whose fiber is a contact manifold

Abstract: It is shown that Maxwell's equations in media without source can be written as a contact Hamiltonian vector field restricted to a Legendre submanifold, where this submanifold is in a fiber space of a bundle and is generated by either electromagnetic energy functional or co-energy functional. Then, it turns out that Legendre duality for this system gives the induction oriented formulation of Maxwell's equations and field intensity oriented one. Also, information geometry of the Maxwell fields is introduced and discussed.

Low-dimensional projective manifolds with nef tangent bundle in positive characteristic

Abstract: We classify smooth projective varieties with nef tangent bundle in positive characteristic, when the varieties are surfaces or Fano 3-folds. Furthermore, some related problems will be discussed.

Equidistribution of closed geodesics along random walk trajectories with respect to the harmonic invariant measure

Abstract: We prove that for suitable random walks on isometry groups of $CAT(-1)$ spaces, typical sample paths eventually land on loxodromic elements which equidistribute with respect to a flow invariant measure on the unit tangent bundle of the quotient space.

A lower bound for the volumes of complements of periodic geodesics

Abstract: Every closed geodesic $\gamma$ on a surface has a canonically associated knot $\widehat\gamma$ in the projective unit tangent bundle. We study, for $\gamma$ filling, the volume of the associated knot complement with respect to its unique complete hyperbolic metric. We provide a lower bound for the volume relative to the number of homotopy classes of $\gamma$-arcs in each pair of pants of a pants decomposition of the surface.

Legendre curves and singularities of a ruled surface according to rotation minimizing frame

Abstract: In this paper, Legendre curves on unit tangent bundle are given using rotation minimizing (RM) vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classifed.

On the stability of tangent bundle on double coverings

Abstract: Let Y be a smooth projective surface defined over an algebraically closed field k with char k ≠ 2, and let τ: X → Y be a double covering branched along a smooth divisor. We show that IX is stable with respect to τ*H if the tangent bundle IY is semi-stable with respect to some ample line bundle H on Y.

Arithmetic, geometry and dynamics in the unit tangent bundle of the modular orbifold

Abstract: 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 $\mathfrak{p}:M\to\mathbb{H}/\text{PSL}(2,\mathbb Z)$ 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 $y\to0$: for every smooth function $f:M\to \mathbb R$ with compact support $m_y(f)=m_0(f)+o(y^\frac12)$ as $y\to0$. 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.

Characterization of the Unit Tangent Sphere Bundle with $ g $-Natural Metric and Almost Contact B-metric Structure

Abstract: We consider unit tangent sphere bundle of a Riemannian manifold $ (M,g) $ as a $ (2n+1) $-dimensional manifold and we equip it with pseudo-Riemannian $ g $-natural almost contact B-metric structure. Then, by computing coefficients of the structure tensor $ F$, we completely characterize the unit tangent sphere bundle equipped to this structure, with respect to the relevant classification of almost contact B-metric structures, and determine a class such that the unit tangent sphere bundle with mentioned structure belongs to it. Also, we find some curvature conditions such that the mentioned structure satisfies each of eleven basic classes.

Unit tangent sphere bundles with the Reeb flow invariant Ricci operator

Abstract: In this paper, we study unit tangent sphere bundles T1M whose Ricci operator $\bar{S}$ is Reeb flow invariant, that is, Lξ$\bar{S}$ = 0. We prove that for a 3-dimensional Riemannian manifold M, T1M satisfies Lξ$\bar{S}$ = 0 if and only if M is of constant curvature 1. Also, we prove that for a 4-dimensional Riemannian manifold M, T1M satisfies Lξ $\bar{S}$ = 0 and l$\bar{S}$ξ = 0 if and only if M is of constant curvature 1 or 2, where l = $\bar{R}$(·,ξ)ξ is the characteristic Jacobi operator.

Volumes of hyperbolic three-manifolds associated to modular links

Abstract: Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a hyperbolic $3$-manifold, which thus 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

Some applications on tangent bundle with Kaluza-Klein metric

Abstract: In this paper, differential equations of geodesics; parall elism, incompressibility and closeness conditions of the h orizontal and complete lift of the vector fields are investigated with r espect to Kaluza-Klein metric on tangent bundle.

A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank

Abstract: We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from a Morin map. When the tangent distribution is the contact structure, we characterize singularities of the bundle homomorphism by using the Hamilton vector fields.

Stable Strata of Geodesics in Outer Space

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

(1,1)-Tensor sphere bundle of Cheeger–Gromoll type

Abstract: We construct a metrical framed $$f(3,-1)$$ -structure on the (1, 1)-tensor bundle of a Riemannian manifold equipped with a Cheeger–Gromoll type metric and by restricting this structure to the (1, 1)-tensor sphere bundle, we obtain an almost metrical paracontact structure on the (1, 1)-tensor sphere bundle. Moreover, we show that the (1, 1)-tensor sphere bundles endowed with the induced metric are never space forms.

G2 fibre bundle structure on an oriented 3-dimensional manifold

Abstract: By using algebraic structures of Clifford algebras and octonions, we show that there exists the G2 principal fibre bundle structure on any oriented 3-dimensional C Riemannian manifold

The Newton–Nelson Equation on Fiber Bundles with Connections

Abstract: The paper is a survey with modifications on the research of the so-called Newton–Nelson equation (the equation of motion in Nelson’s stochastic mechanics) on the total space of a bundle in two cases: where the base of the bundle is a Riemannian manifold and the bundle is real and where the base of the bundle is a Lorentz manifold and the bundle is complex. In the latter case, we describe the relations with the equation of motion of the quantum particle in the classical gauge field (the above-mentioned connection). Moreover, a certain second-order ordinary differential equation on the bundle with connection that is interpreted as the equation of motion of the classical particle in the classical gauge field is described.