Showing papers on "Unit tangent bundle published in 2016"

Lie Sphere Geometry

Abstract: This chapter presents the method of moving frames in Lie sphere geometry. This involves a number of new ideas, beginning with the fact that some Lie sphere transformations are not diffeomorphisms of space S3, but rather of the unit tangent bundle of S3. This we identify with the set of pencils of oriented spheres in S3, which is identified with the set \( \varLambda \) of all lines in the quadric hypersurface \( Q \subset \mathbf{P}(\mathbf{R}^{4,2}) \). The set \( \varLambda \) is a five-dimensional subspace of the Grassmannian G(2, 6). The Lie sphere transformations are the projective transformations of P(R4, 2) that send Q to Q. This is a Lie group acting transitively on \( \varLambda \). The Lie sphere transformations taking points of S3 to points of S3 are exactly the Mobius transformations, which form a proper subgroup of the Lie sphere group. In particular, the isometry groups of the space forms are natural subgroups of the Lie sphere group. There is a contact structure on \( \varLambda \) invariant under the Lie sphere group. A surface immersed in a space form with a unit normal vector field has an equivariant Legendre lift into \( \varLambda \). A surface conformally immersed into Mobius space with an oriented tangent sphere map has an equivariant Legendre lift into \( \varLambda \). This chapter studies Legendre immersions of surfaces into this homogeneous space \( \varLambda \) under the action of the Lie sphere group. A major application is a proof that all Dupin immersions of surfaces in a space form are Lie sphere congruent to each other.

Involutes of fronts in the Euclidean plane

Abstract: For a regular plane curve, an involute of it is the trajectory described by the end of a stretched string unwinding from a point of the curve. Even for a regular curve, the involute always has a singularity. By using a moving frame along the front and the curvature of the Legendre immersion in the unit tangent bundle, we define an involute of the front in the Euclidean plane and give properties of it. We also consider a relationship between evolutes and involutes of fronts without inflection points. As a result, the evolutes and the involutes of fronts without inflection points are corresponding to the differential and the integral of the curvature of the Legendre immersion.

Symmetric differentials on complex hyperbolic manifolds with cusps

Abstract: Let $(X, D)$ be a logarithmic pair, and let $h$ be a singular metric on the tangent bundle, smooth on the open part of $X$. We give sufficient conditions on the curvature of $h$ for the logarithmic and the standard cotangent bundles to be big. As an application, we give a metric proof of the bigness of logarithmic cotangent bundle on any toroidal compactification of a bounded symmetric domain. Then, we use this singular metric approach to study the bigness and the nefness of the standard tangent bundle in the more specific case of the ball. We obtain effective ramification orders for a cover $X' \longrightarrow X$, etale outside the boundary, to have all its subvarieties with big cotangent bundle. We also prove that the standard tangent bundle of such a cover is nef if the ramification is high enough. Moreover, the ramification orders we obtain do not depend on the dimension of the quotient of the ball we consider.

Intersection norms on surfaces and Birkhoff cross sections

Abstract: For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of its integer points. We give an interpretation of these points in terms of certain coorientations of the original collection of curves. Our main result is a classification statement: when the surface has constant curvature and the curves are geodesics, integer points in the interior of the dual unit ball classify isotopy classes of Birkhoff cross sections for the geodesic flow (on the unit tangent bundle to the surface) whose boundary is the symmetric lift of the collection of geodesics. Birkhoff cross sections in particular yield open-book decompositions of the unit tangent bundle.

Lift of semi-symmetric non-metric connection on a Kähler manifold

Abstract: The tangent bundle endowed with semi-symmetric non-metric connection obtained by vertical and complete lifts of a semi-symmetric non-metric connection on the base manifold and proposes to study the tangent bundle of Kahler manifold. Finally we obtain some theorems for Nijenhuis tensor on the tangent bundle of a Kahler manifold.

Integrable Systems with Variable Dissipation on the Tangent Bundle of a Sphere

Abstract: Many problems of multidimensional dynamics involve systems for which the spaces of states are spheres of finite dimension and the spaces of phases are the tangent bundles of such spheres. We study conservative systems and present nonconservative force fields such that the systems involving such forces possess a complete collection of first integrals that are expressed through a finite combination of elementary functions and, in general, are transcendental functions of their variables. Bibliography: 32 titles.

Logarithm Laws for Equilibrium States in Negative Curvature

Abstract: Let M be a pinched negatively curved Riemannian manifold, whose unit tangent bundle is endowed with a Gibbs measure mF associated with a potential F. We compute the Hausdorff dimension of the conditional measures of mF. We study the mF-almost sure asymptotic penetration behaviour of locally geodesic lines of M into small neighbourhoods of closed geodesics, and of other compact (locally) convex subsets of M. We prove Khintchine-type and logarithm law-type results for the spiraling of geodesic lines around these objects. As an arithmetic consequence, we give almost sure Diophantine approximation results of real numbers by quadratic irrationals with respect to general Holder quasi-invariant measures.

Self-adjoint extensions of differential operators on Riemannian manifolds

Abstract: We study \(H=D^*D+V\), where D is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold M, and V is a Hermitian bundle endomorphism. In the case when M is geodesically complete, we establish the essential self-adjointness of positive integer powers of H. In the case when M is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of H, expressed in terms of the behavior of V relative to the Cauchy boundary of M.

Curvature restrictions on a manifold with a flat Higgs bundle

Abstract: We shall prove a semi-negative curvature property for a manifold with a flat admissible Higgs bundle.

Tangent Bundle of the Hypersurfaces in a Euclidean Space

Abstract: We consider an immersed orientable hypersurface f : M ! R n+1 of the Euclidean space (f an immersion), and observe that the tan- gent bundle TM of the hypersurface M is an immersed submanifold of the Euclidean space R 2n+2 . Then we show that in general the induced metric on TM is not a natural metric and obtain expressions for the horizontal and vertical lifts of the vector fields on M. We also study the special case in which the induced metric on TM becomes a natural metric and show that in this case the tangent bundle TM is trivial.

The Lagrange-Poincaré equations for a mechanical system with symmetry on the principal fiber bundle over the base represented by the bundle space of the associated bundle

Abstract: The Lagrange--Poincare equations for a mechanical system which describes the interaction of two scalar particles that move on a special Riemannian manifold, consisting of the product of two manifolds, the total space of a principal fiber bundle and the vector space, are obtained. The derivation of equations is performed by using the variational principle developed by Poincare for the mechanical systems with a symmetry. The obtained equations are written in terms of the dependent variables which, as in gauge theories, are implicitly determined by means of equations representing the local sections of the principal fiber bundle.

Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

Abstract: We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathfrak{F})$ of a compact minimal lamination $(M,\mathfrak{F})$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal and examples where this action is not minimal. In the first case, we prove that if $\mathfrak{F}$ has a leaf which is not simply connected, the horocyle flow is topologically transitive.

H-contact unit tangent sphere bundles of Riemannian manifolds

Abstract: A contact metric manifold is said to be H-contact, if its characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold M equipped with the standard contact metric structure is H-contact if and only if M is 2-stein.

N-Legendre and N-Slant Curves in the Unit Tangent Bundle of Minkowski Surfaces

Abstract: Let $(\mathbb{M}_{1}^{2},g)$ be a Minkowski surface and $(T_1\mathbb{M}_1^2, g_1)$ its unit tangent bundle endowed with the pseudo-Riemannian induced Sasaki metric. We extend in this paper the study of the N-Legendre and N-slant curves which the inner product of normal vector and Reeb vector is zero and nonzero constant respectively in $\left( T_1 \mathbb{M}_1^2, g_1 \right)$, given in \cite{hmy}, to the Minkowski context and several important characterizations of these curves are given. ewline

Canonical contact unit cotangent bundle

Abstract: We describe an explicit open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of a compact surface.

Model spaces in sub-Riemannian geometry

Abstract: We consider sub-Riemannian spaces admitting an isometry group that is maximal in the sense that any linear isometry between the horizontal tangent spaces is realized by a global isometry We will show that these spaces have a canonical choice of partial connection on their horizontal bundle, which is determined by isometries and generalizes the Levi-Civita connection for the special case of Riemannian model spaces The number of invariants needed to describe model spaces with the same tangent cone is in general greater than one, and these invariants are not necessarily related to the holonomy of the canonical connections

A study on fuzzy tangent bundle of fuzzy Banach manifold

Abstract: In this paper, we study some of fuzzy topological and analytical properties of fuzzy tangent bundle and fuzzy cotangent bundle of fuzzy Banach manifold.

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.

Curvature-dependent energies minimizers and visual curve completion

Abstract: Geometrical actions often used to describe elastic properties of elastic rods and fluid membranes have been proposed recently to explain functional mechanism of the primary visual cortex V1. These energies are defined in terms of functionals depending on the Frenet–Serret curvatures of a curve (profile curve, for axisymmetric membranes) and are relevant in image restoration by curve completion. In this context, extremals of length, total squared curvature (bending energy) and total squared torsion, acting on spaces of curves of the unit tangent bundle of the plane, are studied here. We first see that Sub-Riemannian geodesics in $${\mathbb {R}}^2\times {\mathbb {S}}^1$$ project down to minimizers of a total curvature type energy in the plane. This motivates us to analyze the associated variational problem in Euclidean space under different boundary conditions. Although, as we show, parametrized extremals can be obtained by quadratures, their concrete explicit determination faces technical difficulties which can be overcome numerically. We use a numerical approach, based on a gradient descent method, to determine both critical trajectories for these three energies and their projection into the image plane under different boundary and isoperimetric constraints.

Totally Null Surfaces in Neutral K¨ahler 4-Manifolds

Abstract: We study the totally null surfaces of the neutral Kahler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (α-planes) or anti-self-dual (β-planes) and so we consider α-surfaces and β-surfaces. The metric of the examples we study, which include the spaces of oriented geodesics of 3-manifolds of constant curvature, are anti-self-dual, and so it is well-known that the α-planes are integrable and α-surfaces exist. These are holomorphic Lagrangian surfaces, which for the geodesic spaces correspond to totally umbilic foliations of the underlying 3-manifold. The β-surfaces are less known and our interest is mainly in their description. In particular, we classify the β-surfaces of the neutral Kahler metric on TN, the tangent bundle to a Riemannian 2-manifold N. These include the spaces of oriented geodesics in Euclidean and Lorentz 3-space, for which we show that the β-surfaces are affine tangent bundles to curves of constant geodesic curvature on S2 and H2, respectively. In addition, we construct the β-surfaces of the space of oriented geodesics of hyperbolic 3-space.

Fano n-folds with nef tangent bundle and Picard number greater than n-5

Abstract: We prove that Fano n-folds with nef tangent bundle and Picard number greater than $$n-5$$ are rational homogeneous manifolds.

Second order time dependent tangent bundles and their applications

Abstract: The aim of this paper is to geometrize time dependent Lagrangian mechanics in a way that the framework of second order tangent bundles plays an essential role. To this end, we first introduce the concepts of time dependent connections and time dependent semisprays on a manifold $M$ and their induced vector bundle structures on the second order time dependent tangent bundle $\R\times T^2M$. Then we turn our attention to regular time Lagrangians and their interaction with $\R\times T^2M$ in different situations such as mechanical systems with potential fields, external forces and holonomic constraints. Finally we propose an examples to support our theory.

CR-structures of codimension 2 on tangent bundles in Riemann–Finsler geometry

Abstract: We determine a 2-codimensional CR-structure on the slit tangent bundle $$T_0M$$ of a Finsler manifold (M, F) by imposing a condition on the almost complex structure $$\Psi $$ associated to F when restricted to the structural distribution of a framed f-structure. This condition is satisfied when (M, F) is of scalar flag curvature (particularly flat). In the Riemannian case (M, g) this last condition means that g is of constant curvature. This CR-structure is finally generalized by using one positive parameter but under more difficult conditions.

The unit tangent sphere bundle whose characteristic jacobi operator is pseudo-parallel

Abstract: We study the characteristic Jacobi operator l = R̄(·, ξ)ξ (along the Reeb flow ξ) on the unit tangent sphere bundle T1M over a Riemannian manifold (Mn, g). We prove that if l is pseudo-parallel, i.e., R̄ · l = LQ(ḡ, l), by a non-positive function L, then M is locally flat. Moreover, when L is a constant and n 6= 16, M is of constant curvature 0 or 1.

On the second-order tangent bundle with horizontal lift connection

Abstract: Let M be an n–dimensional differentiable manifold equipped with a torsion-free linear connection ∇ and T2 M be its second-order tangent bundle. The present paper aims to study some curvature properties of T2 M with respect to the horizontal lift connection H∇.

Weakly symmetry of a class of g-natural metrics on tangent bundles

Abstract: The paper considers a class of g-natural metrics G on the tangent bundle TM of a Riemannian manifold (M, g). We prove that the flatness of g is necessary and sufficient for a metric G to be weakly symmetric (recurrent or pseudo-symmetric). Also, it is shown that the weak symmetry and recurrent or pseudo-symmetry properties of Sasakian lift metric, studied by Bejan and Crasmareanu, and Binh and Tamassy, respectively, are special cases of our result.

The ciconia metric on the tangent bundle of an almost-Hermitian manifold

Abstract: 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.

Intersection norms on surfaces and Birkhoff surfaces for geodesic flows

Abstract: For every finite collection of curves on a surface, we define an associated (semi-)norm on the first homology group of the surface. The unit ball of the dual norm is the convex hull of finitely many integer points. We give an interpretation of these points in terms of certain coorien-tations of the original collection of curves. Our main result is a classification statement: when the surface has constant curvature and the curves are geodesics, integer points in the interior of the unit ball of the dual norm classify isotopy classes of Birkhoff sections for the geodesic flow (on the unit tangent bundle to the surface) whose boundary is the symmetric lift of the collection of geodesics. These Birkhoff sections also yield numerous open-book decompositions of the unit tangent bundle.

On the $k$-Semispray of Nonlinear Connections in $k$-Tangent Bundle Geometry

Abstract: In this paper we present a method by which is obtained a sequence of $k$-semisprays and two sequences of nonlinear connections on the $k$-tangent bundle $T^kM$, starting from a given one. Interesting particular cases appear for Lagrange and Finsler spaces of order $k$.

Adapted metrics and Webster curvature in Finslerian 2-dimensional geometry

Abstract: The Webster scalar curvature is computed for the sphere bundle T1S of a Finsler surface (S, F) subject to the Chern-Hamilton notion of adapted metrics. As an application, it is derived that in this setting (T1S, gSasaki) is a Sasakian manifold homothetic with a generalized Berger sphere, and that a natural Cartan structure is arising from the horizontal 1-forms and the author associates a non-Einstein pseudo-Hermitian structure. Also, one studies when the Sasaki type metric of T1S is generally adapted to the natural co-frame provided by the Finsler structure.