Showing papers on "Unit tangent bundle published in 2004"

On differentiability of SRB states for partially hyperbolic systems

Abstract: Consider a one parameter family of diffeomorphisms f e such that f 0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties Let νe be any u-Gibbs state for f e We prove (Theorem 1) that for any C ∞ function A the map e→νe(A) is differentiable at e=0 This implies (Corollary 22) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to e We apply this result (Corollary 33) to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique SRB measure with good statistical properties

Principe variationnel et groupes Kleiniens

Abstract: Let Γ be a nonelementary Kleinian group acting on a Cartan-Hadamard manifold $\tilde{X}$; denote by Λ(Γ) the nonwandering set of the geodesic flow (φt) acting on the unit tangent bundle T1($\tilde{X}$/Γ). When Γ is convex cocompact (i.e., Λ(Γ) is compact), the restriction of (φt) to Λ(Γ) is an Axiom A flow: therefore, by a theorem of Bowen and Ruelle, there exists a unique invariant measure on Λ(Γ) which has maximal entropy. In this paper, we study the case of an arbitrary Kleinian group Γ. We show that there exists a measure of maximal entropy for the restriction of(φt) to Λ(Γ) if and only if the Patterson-Sullivan measure is finite; furthermore when this measure is finite, it is the unique measure of maximal entropy. By a theorem of Handel and Kitchens, the supremum of the measure-theoretic entropies equals the infimum of the entropies of the distances d on Λ(X); when Γ is geometrically finite, we show that this infimum is achieved by the Riemannian distance d on Λ(X).

Note on the classification theorems of $g$-natural metrics on the tangent bundle of a Riemannian manifold $(M,g)$

Abstract: In (7), it is proved that all g-natural metrics on tangent bundles of m-dimen- sional Riemannian manifolds depend on arbitrary smooth functions on positive real numbers, whose number depends on m and on the assumption that the base manifold is oriented, or non-oriented, respectively. The result was originally stated in (8) for the oriented case, but the smoothness was assumed and not explicitly proved. In this note, we shall prove that, both in the oriented and non-oriented cases, the functions generating the g-natural metrics are, in fact, smooth on the set of all nonnegative real numbers.

On Manifolds Whose Tangent Bundle is Big and 1-Ample

Abstract: A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample. The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ?almost? ample, the present result is yet another extension of the celebrated Mori paper ?Projective manifolds with ample tangent bundles? (Ann. of Math. 110 (1979) 593?606). The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results.

Some Classes of Almost Anti-Hermitian Structures on the Tangent Bundle

Abstract: In [11] we have considered a family of natural almost anti-Hermitian structures (G, J) on the tangent bundle TM of a Riemannian manifold (M, g), where the semi-Riemannian metric G is a lift of natural type of g to TM, such that the vertical and horizontal distributions VTM, HTM are maximally isotropic and the almost complex structure J is a usual natural lift of g of diagonal type interchanging VTM, HTM (see [5], [15]). We have obtained the conditions under which this almost anti-Hermitian structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification given in [1]. In this paper we consider another semi-Riemannian metric G on TM such that the vertical and horizontal distributions are orthogonal to each other. We study the conditions under which the above almost complex structure J defines, together with G, an almost anti-Hermitian structure on TM. Next, we obtain the conditions under which this structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification in [1].

On manifolds with trivial logarithmic tangent bundle

Abstract: By a classical result of Wang [14] a connected compact complex manifold X has holomorphically trivial tangent bundle if and only if there is a connected complex Lie group G and a discrete subgroup Γ such that X is biholomorphic to the quotient manifold G/Γ. In particular X is homogeneous. If X is Kähler, G must be commutative and the quotient manifold G/Γ is a compact complex torus. The purpose of this note is to generalize this result to the noncompact Kähler case. Evidently, for arbitrary non-compact complex manifold such a result can not hold. For instance, every domain over C has trivial tangent bundle, but many domains have no automorphisms. So we consider the “open case” in the sense of Iitaka ([7]), i.e. we consider manifolds which can be compactified by adding a divisor. Following a suggestion of the referee, instead of only considering Kähler manifolds we consider manifolds in class C as introduced in [5]. A compact complex manifold X is said to be class in C if there is a surjective holomorphic map from a compact Kähler manifold onto X. Equivalently, X is bimeromorphic to a Kähler manifold ([13]). For example, every Moishezon manifold is in class C. We obtain the following characterization:

Corrected energy of distributions on riemannian manifolds

Abstract: In the mathematical literature there are several functionals which let us measure how the vector fields defined over any Riemannian manifold are ordered. We can ask ourselves which are the optimal vector fields. In fact, we try to measure how far from being parallel our vector field is. We can also extend this question to distributions. Gluck, Ziller [5] and Johnson [6], among others, studied the volume of unit vector fields. They define the volume of a unit vector field to be the volume of the submanifold in the unit tangent bundle defined by ( ). For this, we regard the vector field as a map : → 1 and in 1 we consider the Sasaki metric. We know [5] that in the ambient manifold S3 the Hopf vector fields, and no others, minimize this functional. For higher dimensional spheres, we know [6] that the Hopf vector fields are unstable critical points; that is, they are not even local minima. Wiegmink [8] defined the total bending of a unit vector field . This functional is related to the energy of the map : → 1 , as we shall see in Section 3. Brito [1] proved that the Hopf vector fields in S3 are the only minima of the total bending. Furthermore, he proved a more general result giving an absolute minimum in any dimension of the total bending corrected by the second fundamental form of the orthogonal distribution to the field . The coefficient of this correction vanishes in dimension 3 and then the corrected total bending agrees with the total bending. Similarly to the situation for vector fields, the energy of a -distribution V in a compact oriented Riemannian manifold is the energy of the section of the Grassmann manifold of -planes in induced by V . In this paper, we add to the energy the norms of the mean curvatures of V and its orthogonal distribution (with different weights) introducing in this way the corrected energy. In Theorem 1, we find a lower bound for the corrected energy of a foliation, and

On the volume of unit vector fields on spaces of constant sectional curvature

Abstract: A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima.

Differential and Geometric Structure for the Tangent Bundle of a Projective Limit Manifold.

Abstract: The tangent bundle of a wide class of Frechet manifolds is studied he- re. A vector bundle structure is obtained with structural group a topological subgroup of the general linear group of the fiber type. Moreover, basic geo- metric results, known form the classical case of finite dimensional manifolds, are recovered here: Connections can be defined and are characterized by a generalized type of Christoffel symbols while, at the same time, parallel di- splacements of curves are possible despite the problems concerning differen- tial equations in Frechet spaces.

The magnetic geodesic flow in a homogeneous field on the complex projective space.

Abstract: We prove commutative integrability of the Hamilton system on the tangent bundle of the complex projective space whose Hamiltonian coincides with the Hamiltonian of the geodesic flow and the Poisson bracket deforms due to addition of the Fubini–Study form to the standard symplectic form.

Packing dimensions, transversal mappings and geodesic flows

Abstract: In this work we first generalize the projection result by K. Falconer and J. Howroyd concerning packing dimensions of projected measures on R" to parametrized families of transversal mappings between smooth manifolds and measures on them. After this we compute the packing dimension of the natural projection of a probability measure which is invariant under the geodesic flow on the unit tangent bundle of a two-dimensional Riemannian manifold.

Twistor bundle theory and its application

Abstract: Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm = SO(2m)/U(m)↪ J +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ 1 onJ +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.

A class of Poisson–Nijenhuis structures on a tangent bundle

Abstract: Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson–Nijenhuis structure from a given type (1, 1) tensor field J on Q. It is argued that the complete lift Jc of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to T*Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle, in the case when Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.

Quantum states from tangent vectors

Abstract: We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite-dimensional, compact, classical phase spaces $\mathcal{C}$, by explicitly constructing Hilbert-space vector bundles over ${\mathcal C}$. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle $T({\mathcal C})$, plus a complex line bundle $N({\mathcal C})$. Quantum states (except the vacuum) appear as tangent vectors to ${\mathcal C}$. The vacuum state appears as the fibrewise generator of $N({\mathcal C})$. Holomorphic line bundles $N({\mathcal C})$ are classified by the elements of ${\rm Pic}({\mathcal C})$, the Picard group of ${\mathcal C}$. In this way ${\rm Pic}({\mathcal C})$ appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case when ${\mathcal C}$ is complex projective space CPn.

Some geometric structures on the tangent bundle of a Riemannian manifold

Abstract: It is defined a new almost complex structure with Norden metric (hyperbolic metric) on the tangent bundle TM of an n—dimensional Riemannian manifold M. Next, the conditions under which the considered almost complex structure with Norden metric belongs to one of the eight classes of almost complex manifolds with Norden metric obtained by G. T. Ganchev and D. V. Borisov in the classification from [2] there are studied.

The natural functions on the cotangent bundle of higher order vector tangent bundles over fibered manifolds

Abstract: For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T *(J (r,s,q)(Y, R 1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T *(J (r,s) (Y, R)0)*→R for any Y as above.

The tangent Lie group bundle and the tangent groupoid of a Heisenberg manifold

Abstract: In this paper we study the tangent structure of a Heisenberg manifold (M, H) in terms of a tangent Lie group bundle GM and a tangent groupoid GHM canonically associated to it via a canonical Levi form which encodes the Lie group structure. It is interesting to look at the tangent Lie group bundle in the main examples of Heisenberg manifolds (Heisenberg group, foliations, contact and CR manifolds) because we can show that these examples can be characterized by means of their tangent Lie group bundle. Concerning the tangent groupoid we get a complete analogue in the Heisenberg setting of Connes' tangent groupoid of a manifold. In particular, as the latter the former is a differentiable groupoid in the category of manifolds with boundary.

On adapted bi-conformal metrics

Abstract: The geometry of the tangent bundle is used to define a particular class of metrics called adapted bi-conformal. These metrics are defined and studied on the holomorphic cotangent bundle of a Kahler manifold. Kahlerian adapted bi-conformal metrics are totally classified and their curvature expressed. The Eguchi-Hanson metric appears as a particular example.

Flow invariant subsets for geodesic flows of manifolds with non-positive curvature

Abstract: Consider a closed, smooth manifold M of non-positive curvature. Write p:UM→M for the unit tangent bundle over M and let ℛ > denote the subset consisting of all vectors of higher rank. This subset is closed and invariant under the geodesic flow φ on UM. We define the structured dimension s-dim ℛ > which, essentially, is the dimension of the set p(ℛ > ) of base points of ℛ > . The main result of this paper holds for manifolds with s-dim ℛ > 0, there is an e-dense, flow invariant, closed subset Ξ e ⊂UM∖ℛ > such that p(Ξ e )=M.

Stability of the characteristic vector field of a sasakian manifold

Abstract: T heVolume of a unit vector field is the volume of its image in the unit tangent bundle. On the standard odd-dimensional spheres, the Hopf vector fields - that is, unit vector fields tangent to the fiber of any Hopf fibration - are critical for the volume functional, but they are not always stable. In fact, stability depends on the radius r of the sphere : for every odd dimension n there exists a "critical radius" such that, if r is lower than this radius the Hopf fields are stable on S n (r) and conversely. In this article, we show that this phenomenon occurs for the characteristic vector field of any Sasakian manifold. We then derive two invariants of a Sasakian manifold, its E-stability and its stability number.

The unit tangent sphere bundle of a complex space form

Abstract: In this paper, we study the unit tangent sphere bun- dles T1M(4c) of complex space forms M(4c) with constant holomor- phic sectional curvature 4c. In particular, we determine T1M(4c) whose Ricci tensors satisfy the Einstein-like conditions.

Powers of the curvature operator of space forms and geodesics of the tangent bundle

Abstract: It is well known that if Г is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold, then all geodesic curvatures of the projected curve λ=π 1463-01 Г are constant. In this paper, we consider the case of the tangent (sphere) bundle over real, complex, and quaternionic space forms and give a unified proof of the following property: All geodesic curvatures of the projected curve are zero beginning with k3, k6, and k10 for the real, complex, and quaternionic space forms, respectively.

Generalized Doppler effect in spaces with a transport along paths

Abstract: An analog of the classical Doppler effect is investigated in spaces (manifolds) whose tangent bundle is endowed with a transport along paths, which, in particular, can be parallel one. The obtained results are valid irrespectively to the particles mass, i.e. they hold for massless particles (e.g. photons) as well as for massive ones.