Showing papers on "Unit tangent bundle published in 2003"
TL;DR: In this article, the authors study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the ergodic averages of horocycle flows.
Abstract: There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu=f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann surface M of finite area and f is a given function on SM. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the ergodic averages of horocycle flows.
191 citations
TL;DR: In this paper, it was shown that every Stein manifold X of dimension n admits holomorphic functions with pointwise independent differentials, and that this number is maximal for every n. In particular, X admits a holomorphic function without critical points.
Abstract: We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle E in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the tangent bundle of a holomorphic foliation of X. If X is parallelizable, it admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any dimension; the question whether such X also admits a submersion (=immersion) in C^n remains open. Our proof involves a blend of techniques (holomorphic automorphisms of Euclidean spaces, solvability of the di-bar equation with uniform estimates, Thom's jet transversality theorem, Gromov's convex integration method). A result of possible independent interest is a lemma on compositional splitting of biholomorphic mappings close to the identity (Theorem 4.1).
87 citations
TL;DR: In this paper, it was shown that for any constant ν > 0, the Kronecker sequence embedded in T1M along a long closed horocycle becomes equidistributed in T 1M for almost all α,p rovided that � = M ν →∞.
Abstract: It is well known that (i) for every irrational number α the Kronecker sequence mα (m =1 ,...,M ) is equidistributed modulo one in the limit M →∞ , and (ii) closed horocycles of lengthbecome equidis- tributed in the unit tangent bundle T1M of a hyperbolic surface M of finite area, as � →∞ . In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant ν> 0 the Kronecker sequence embedded in T1M along a long closed horocycle becomes equidistributed in T1M for almost all α ,p rovided that � = M ν →∞ . This equidistribution result holds in fact under explicit diophantine conditions on α (e.g. for α = √ 2) provided that ν< 1, or ν< 2 with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for ν =2 , our equidistribution theorem implies a recent result of Rudnick and Sar- nak on the uniformity of the pair correlation density of the sequence n 2 α modulo one.
48 citations
TL;DR: In this article, the information dimension of a probability measure on the unit tangent bundle of a Riemannian manifold is defined as the probability of the measure being invariant under the geodesic flow on the manifold.
Abstract: Let M be a compact Riemannian surface (a two-dimensional Riemannian manifold),with μ a probability measure on the unit tangent bundle SM invariant under the geodesic flow. We are interested in understanding the image, and especially the dimension of the image, of μ under the natural projection SM →M . It will expire that the properties of interest of this specific projection are immediate consequences of general methods used to study properties of a typical member of a family of projections. Many authors contributed in this directions, and we make no attempt to be thorough. We refer the reader to [9],[6],[5], [10],[11], [4], [14] and the recent [13], where additional references can also be found regarding the dimension of projections. There are many different possible ways to define the dimension of a measure μ. In what follows, we shall use the information dimension, which for the measures we will consider, namely measures invariant under the geodesic flow on the unit tangent bundle of a surface, is closely related through a theorem of L-S. Young [17] to the entropy (we also mention the related [3] by M. Brin and A. Katok). It is defined as follows:
30 citations
TL;DR: In this article, the authors define a class of lengths of paths in a sub-Riemannian manifold, which includes the length of horizontal paths and transverse paths, by integrating an infinitesimal measure which generalizes the norm on the tangent space.
Abstract: We define a class of lengths of paths in a sub-Riemannian manifold. It includes the length of horizontal paths but also measures the length of transverse paths. It is obtained by integrating an infinitesimal measure which generalizes the norm on the tangent space. This requires the definition and the study of the metric tangent space (in Gromov's sense). As an example, we compute those measures in the case of contact sub-Riemannian manifolds.
27 citations
TL;DR: In this article, the authors dealt with properties of the Cheeger-Gromoll metric g introduced by Musso and Tricerri in 1988 and showed that it is non-rigid.
Abstract: This paper deals with properties of the Cheeger-Gromoll metric g introduced in 1988 by Musso and Tricerri on the tangent bundle TM associated to a given Riemannian metric ðM; gÞ. One can find here essentially the two following results: 1. A classification of Killing vector fields on ðTM; gÞ 2. A generalization of a result of M. Sekizawa concerning the non rigidity of the Cheeger-Gromoll metric.
25 citations
Posted Content•
TL;DR: In this article, a non-Euclidean analysis on Lie groups endowed with left invariant distributions, seen as sub-Riemannian manifolds, is presented.
Abstract: This paper is about non-Euclidean analysis on Lie groups endowed with left invariant distributions, seen as sub-Riemannian manifolds.
This is a an updated version, which will be modified according to the contributions of the other participants to the Borel Seminar. The current version can be downloaded at this http URL . All comments are welcomed.
19 citations
TL;DR: In this paper, a generalization of Lagrangian geometry is proposed, which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangians by a family of compatible, local, Lagrangeian functions.
Abstract: Lagrange geometry is the geometry of the tensor field defined by
the fiberwise Hessian of a nondegenerate Lagrangian function on
the total space of a tangent bundle. Finsler geometry is the
geometrically most interesting case of Lagrange geometry. In this
paper, we study a generalization which consists of replacing the
tangent bundle by a general tangent manifold, and the Lagrangian
by a family of compatible, local, Lagrangian functions. We give
several examples and find the cohomological obstructions to
globalization. Then, we extend the connections used in Finsler
and Lagrange geometry, while giving an index-free presentation of
these connections.
19 citations
TL;DR: In this paper, a geodesic γ on the unit tangent sphere bundle T 1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it.
Abstract: A geodesic γ on the unit tangent sphere bundle T1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it. We study the curves x. In particular, we investigate for which manifolds (M, g) all these curves have constant first curvature κ1 or have vanishing curvature κi for some i = 1, 2 or 3.
13 citations
TL;DR: The Hopf vector field is unique among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodeic in the unit tangent bundle with Sasaki metric.
Abstract: We prove that the Hopf vector field is unique among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric As an application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field
10 citations
TL;DR: In this paper, it was shown that the total bending along a geodesic arc on the boundary of the convex core is bounded by a function of the length of the arc.
Abstract: In this paper we consider hyperbolic manifolds with incompressible convex core boundary. We show that total bending along a geodesic arc on the boundary of the convex core is bounded above by a function of its length. Integrating this function over the unit tangent bundle of the boundary of the convex core we obtain a new universal upper bound on the total bending of the convex core boundary. Furthermore, we produce a new universal upper bound on the lipschitz constant for the map from the convex core boundary to the hyperbolic structure at infinity. These results improve on earlier bounds of Bridgeman and Canary. Let N = H/Γ be an orientable hyperbolic manifold with domain of discontinuity Ω(Γ) and limit set LΓ. In this paper we restrict ourselves to the case when all the components of Ω(Γ) are simply connected. This is a natural restriction to make and includes the set of quasi-fuchsian groups. Let CH(LΓ) be the convex hull of Γ and βΓ be the bending lamination on ∂CH(LΓ). Let C(N) = CH(LΓ)/Γ be the convex core, and βN be the bending lamination on ∂C(N). Then we observe that ∂C(N) is incompressible if and only if the components of Ω(Γ) are all simply connected. If α is a geodesic arc in CH(LΓ), the average bending B(α) is defined to be the bending per unit length, or specifically
TL;DR: In this paper, the authors study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold and understand implications of properties of interest in partial differential equations.
Abstract: We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold The aim is to understand implications of properties of interest in partial differential equations
TL;DR: In this article, it was shown that the Pesin set of an expansive geodesic flow in compact manifold with no conjugate points and bounded asymptote coincides with an open and dense set of the unit tangent bundle.
Abstract: In this paper, we show that the Pesin set of an expansive geodesic flow in compact manifold with no conjugate points and bounded asymptote coincides a.e with an open and dense set of the unit tangent bundle. We also show that the set of hyperbolic periodic orbits is dense in the unit tangent bundle.
TL;DR: In this paper, the authors determine all Riemannian manifolds for which the tangent sphere bundles, equipped with the Sasaki metric, are local or global product manifolds.
Abstract: We determine all Riemannian manifolds for which the tangent sphere bundles, equipped with the Sasaki metric, are local or global Riemannian product manifolds.
Posted Content•
TL;DR: In this article, the displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced, and in case these spaces are equipped with a linear connection, the deviation equations (between arbitrary, geodesic or not, paths) in such spaces are investigated.
Abstract: The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between arbitrary, geodesic or not, paths) in such spaces are investigated.
TL;DR: For the Riemannian manifold Mn, two special connections are constructed on the sum of the tangent bundle TMn and the trivial one-dimensional bundle as discussed by the authors, and these connections are flat if and only if the space Mn has a constant sectional curvature ± 1.
Abstract: For the Riemannian manifold Mn two special connections are constructed on the sum of the tangent bundle TMn and the trivial one-dimensional bundle. These connections are flat if and only if the space Mn has a constant sectional curvature ±1. The geometric explanation of this property is given. This construction gives a coordinate-free many-dimensional generalization of the Sasaki connection (Sasaki R 1979 Soliton equations and pseudospherical surfaces Nucl. Phys. B 154 343–57). It is shown that these connections have a close relation to the imbedding of Mn into Euclidean or pseudo-Euclidean (n + 1)-dimension spaces.
Journal Article•
TL;DR: For a compact Riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism as mentioned in this paper.
Abstract: For a compact Riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the Riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S , we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Holder distributions for the geodesic foliation.
Journal Article•
TL;DR: In this article, the authors studied a one-to-one correspondence between DS 2 and a non cylindrical ruled surface and showed that for a restriction of an anti-symmetric linear vector field A along a spherical curve α(t) there exists a non-cylindrical rule surface which corresponds to α (t).
Abstract: T S 2 is a differentiable manifold of dimension 4. For every X ∈ T S 2 , if we set X = (p, x) we have < < p, x >= 0 since p is orthogonal to T p S 2 , therefore p = 1. Those there could exist a one-to-one correspondence between T S 2 and DS 2. In this paper we gave and studied a one-to-one correspondence among T S 2 , DS 2 and a non cylindrical ruled surface. We showed that for a restriction of an anti-symmetric linear vector field A along a spherical curve α(t) there exists a non-cylindrical ruled surface which corresponds to α(t) and has the following parametrization α(t, λ) = α(t) + A(α(t)) + λα(t) So it is possible to study non-cylindrical ruled surfaces as the set of (α(t), A(α(t))), where α(t) ∈ S 2 and A is an anti-symmetric linear vector field in R 3. Key words: dual unit sphere, non-cylindrical ruled surface, spherical curve, anti-symmetric linear vector field, tangent bundle 1. Anti-symmetric linear vector fields Let A = [a ij ] be a fixed real n × n matrix. For each such A we construct a vector field T A on R n by taking its value at each point x ∈ R n to be the negative of the result of applying the matrix A to the vector X, i.e. Definition 1. A vector field T A is called linear vector field ([3]). If A is an anti-symmetric (symmetric, orthogonal, etc.) matrix then T A is called an anti-symmetric (symmetric, orthogonal, etc.) linear vector field.
TL;DR: In this article, it was shown that the bundles of nonholonomic and semi-holonomic second-order frames of a real or complex manifold M can be obtained as extensions of the bundle F2(M) of secondorder jets of (holomorphic) diffeomorphisms of (Kn, 0) into M, where K=R or C. This is a new proof of a theorem of Libermann.
TL;DR: In this paper, the authors studied Riemannian foliations with complex leaves on Kahler manifolds, where the tensor T, the obstruction to the foliation, is interpreted as a holomorphic section of a certain vector bundle.
Abstract: We study Riemannian foliations with complex leaves on Kahler manifolds. The tensor T, the obstruction to the foliation be totally geodesic, is interpreted as a holomorphic section of a certain vector bundle. This enables us to give classification results when the manifold is compact.
TL;DR: In this article, the integrability of a linear r-th order connection on the tangent bundle is characterized geometrically in the sense of higher order G-structures.
Abstract: For a linear r-th order connection on the tangent bundle we characterize geometrically its integrability in the sense of the theory of higher order G-structures. Our main tool is a bijection between these connections and the principal connections on the r-th order frame bundle and the comparison of the torsions under both approaches.
Posted Content•
TL;DR: In this article, the concepts of relative velocity and acceleration, deviation velocity, acceleration and relative momentum of point particles in spaces (manifolds), the tangent bundle of which is equipped with a transport along paths, are introduced.
Abstract: The concepts of relative velocity and acceleration, deviation velocity and acceleration and relative momentum of point particles in spaces (manifolds), the tangent bundle of which is equipped with a transport along paths, are introduced. If the tangent bundle is endowed also with a metric, it gives rise also to the notion of a relative energy. Certain ties between these quantities are considered. The cases of massless particles and of special relativity are presented in this context.
TL;DR: In this article, the Hirzebruch's Riemann-Roch formula for endomorphism bundles over a compact complex was used to prove that the tangent bundle of a complex surface M of general type admits a nontrivial trace-free deformation, unless M is holomorphically covered by the euclidean ball.
Abstract: Using the Hirzebruch's Riemann-Roch formula for endomorphism bundles over a compact complex two-fold we prove that the tangent bundle of a complex surface M of general type admits a nontrivial trace-free deformation, unless M is holomorphically covered by the euclidean ball. It follows that the tangent bundle of the Mostow-Siu surface, which is a Kahler surface with a negative definite curvature tensor, does have a nontrivial trace-free moduli. Among some other results we also point out a relationship between the Kuranishi obstruction and symmetric holomorphic two tensors on a complex surface.
01 Jan 2003
TL;DR: In this article, it was shown that any Riemann structure (M, g) determines a local conformal Kahler manifold on the tangent bundle (TM,π,M).
Abstract: We prove that any Riemann structure (M, g) determines a local conformal Kahler manifold on the tangent bundle (TM,π ,M). The Finsler structures and Lagrange structures have the same properties.
01 Jan 2003
TL;DR: In this article, the notion of a homogeneous 2-π structure Φ on the tangent bundle was defined and its compatibility with the Riemannian structure G determined by the homogeneous lift (2.1) of Finsler fundamental tensor field was investigated.
Abstract: One defines the notion of a homogeneous 2-π structure Φ on the tangent bundle and one investigates its compatibility with the Riemannian structure G determined by the homogeneous lift (2.1) of Finsler fundamental tensor field. The connections compatible with the pair (G,Φ) are studied, too.
01 Jan 2003
TL;DR: In this paper, it was shown that a Riemannian structure on the total space TM of the tangent bundle (TM, π, M) determines an almost Hermitian structure.
Abstract: One proves that a Riemannian structure 𝔾 on the total space TM of the tangent bundle (TM, π, M) determines an almost Hermitian structure on TM.
09 Apr 2003
TL;DR: In this paper, it was shown that a measure on the unit tangent bundle of S 2n, which is even and invariant under the geodesic flow, is not uniquely determined by its projection to S n.
Abstract: Let S n be the n-sphere of constant positive curvature. For n > 2, we will show that a measure on the unit tangent bundle of S 2n , which is even and invariant under the geodesic flow, is not uniquely determined by its projection to S 2n .
Posted Content•
TL;DR: In this paper, it was shown that a measure on the unit tangent bundle of a constant positive curvature space is not uniquely determined by its projection to the positive curvatures.
Abstract: Let $S^{n}$ be the $n$-sphere of constant positive curvature. For $n \geq 2$, we will show that a measure on the unit tangent bundle of $S^{2n}$, which is even and invariant under the geodesic flow, is not uniquely determined by its projection to $S^{2n}$.