Showing papers on "Unit tangent bundle published in 2010"
TL;DR: In this article, a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is shown to be the equivalent of an Eigenvector.
Abstract: We show that a compact contact Ricci soliton with a potential vector field V collinear with the Reeb vector field, is Einstein. We also show that a homogeneous H-contact gradient Ricci soliton is locally isometric to En+1 × Sn(4). Finally we obtain conditions so that the horizontal and tangential lifts of a vector field on the base manifold may be potential vector fields of a Ricci soliton on the unit tangent bundle.
38 citations
TL;DR: In this article, it was shown that semicalibrated integer multiplicity rectifiable 2-cycles have a unique tangent cone at every point in a Riemannian manifold.
Abstract: Semicalibrated currents in a Riemannian manifold are currents that are calibrated by a comass-1 differential form that is not necessarily closed. This extension of the classical notion of calibrated currents is motivated by important applications in differential geometry such as special Legendrian currents, for example. We prove that semicalibrated integer multiplicity rectifiable 2-cycles have a unique tangent cone at every point. The proof is based on the introduction of a new technique that might be useful for other first-order elliptic problems
25 citations
TL;DR: In this article, the (1 − 1)-correlationship between SO(m+1)-invariant homogeneous metrics on V2Rm + 1 and all so-called g-natural metrics on T1Sm was established.
Abstract: It is well known that the unit tangent sphere bundle T1Sm of the standard sphere Sm can be naturally identified with the Stiefel manifold V2Rm+1=SO(m+1)/SO(m−1). In this paper, we construct the (1–1) correspondence between all SO(m+1)-invariant homogeneous metrics on V2Rm+1 and all so-called g-natural metrics on T1Sm.
22 citations
13 Jun 2010
TL;DR: This paper argues that a suitable abstraction is the unit tangent bundle R2 × S1 and shows that a basic principle of “minimum energy consumption” in this space, namely a minimum length completion, entails desired perceptual properties for the completion in the image plane.
Abstract: The phenomenon of visual curve completion, where the visual system completes the missing part (e.g., due to occlusion) between two contour fragments, is a major problem in perceptual organization research. Previous computational approaches for the shape of the completed curve typically follow formal descriptions of desired, image-based perceptual properties (e.g, minimum total curvature, roundedness, etc.). Unfortunately, however, it is difficult to determine such desired properties psychophysically and indeed there is no consensus in the literature for what they should be. Instead, in this paper we suggest to exploit the fact that curve completion occurs in early vision in order to formalize the problem in a space that explicitly abstracts the primary visual cortex. We first argue that a suitable abstraction is the unit tangent bundle R2 × S1 and then we show that a basic principle of “minimum energy consumption” in this space, namely a minimum length completion, entails desired perceptual properties for the completion in the image plane. We present formal theoretical analysis and numerical solution methods, we show results on natural images and their advantage over existing popular approaches, and we discuss how our theory explains recent findings from the perceptual literature using basic principles only.
18 citations
TL;DR: In this article, a class of g -natural metrics on the tangent bundle of a Finsler manifold which is a generalized version of Sasaki-Matsumoto metric and Miron metric is studied.
Abstract: In this article, we study a class of g -natural metrics on the tangent bundle of a Finsler manifold which is a generalized version of Sasaki–Matsumoto metric and Miron metric. Then, we consider on compatible almost complex structure with together the metric confers to the slit tangent bundle of Finsler manifold and structure of locally conformal almost Kahlerian manifold. We find some conditions under which the slit tangent bundle is locally conformal Kahlerian, Kahlerian, locally Euclidean or the Finsler manifold has scalar or constant flag curvature.
18 citations
TL;DR: In this paper, it was shown that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact, and the converse was also proved for metrics of Kaluza-Klein type.
Abstract: We prove that all g-natural contact metric structures on a two-point homogeneous space are homogeneous contact. The converse is also proved for metrics of Kaluza–Klein type. We also show that if (M,g) is an Einstein manifold and is a Riemannian g-natural metric on T1M of Kaluza–Klein type, then is H-contact if and only if (M,g) is 2-stein, so proving that the main result of Chun et al. [‘H-contact unit tangent sphere bundles of Einstein manifolds’, Q. J. Math., to appear. DOI: 10.1093/qmath/hap025] is invariant under a two-parameter deformation of the standard contact metric structure on T1M. Moreover, we completely characterize Riemannian manifolds admitting two distinct H-contact g-natural contact metric structures, with associated metric of Kaluza–Klein type.
17 citations
TL;DR: In this paper, the authors studied the natural G2 structure on the unit tangent sphere bundle SM of any given orientable Riemannian 4-manifold M, as was discovered in Albuquerque and Salavessa (2009,2010) [9,10].
17 citations
TL;DR: In this article, it was shown that there does not exist any Finsler metric such that this structure became locally symmetric or Einstein manifold, and similar results were obtained on a tube around zero section in the tangent bundle in the case of negative constant flag curvature.
Abstract: In this paper, we obtain a Kahler structure on the tangent bundle of a Finsler manifold of positive constant flag curvature. We show that there does not exist any Finsler metric such that this structure became locally symmetric or Einstein manifold. Similar results are obtained on a tube around zero section in the tangent bundle in the case of a Finsler manifold of negative constant flag curvature.
16 citations
TL;DR: In this article, the authors consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T2 for an arbitrary Riemannian metric.
Abstract: In the present work we consider the behavior of the geodesic flow on the unit tangent bundle of the 2-torus T2 for an arbitrary Riemannian metric. A natural non-negative quantity which measures the complexity of the geodesic flow is the topological entropy. In particular, positive topological entropy implies chaotic behavior on an invariant set in the phase space of positive Hausdorff-dimension (horseshoe). We show that in the case of zero topological entropy the flow has properties similar to integrable systems. In particular, there exists a non-trivial continuous constant of motion which measures the direction of geodesics lifted onto the universal covering \({\mathbb{R}^{2}}\) . Furthermore, those geodesics travel in strips bounded by Euclidean lines. Moreover, we derive necessary and sufficient conditions for vanishing topological entropy involving intersection properties of single geodesics on T2.
15 citations
TL;DR: In this article, a class of complex structures on the generalized tangent bundle of a smooth manifold M endowed with a torsion free linear connection, ∇, is studied and integrability conditions are studied.
13 citations
Posted Content•
TL;DR: In this article, the authors present a new formulation of lifting problems in terms of geometry on the loop space, which explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.
Abstract: We review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.
01 Jan 2010
TL;DR: The main purpose of as mentioned in this paper is to give a construction of mirror dual Calabi-Yau submanifolds inside of a G 2 manifold, which can then be used to define different complex and symplectic structures on certain 6-dimensional subbundles of T(M).
Abstract: The main purpose of this chapter is to give a construction of certain “mirror dual” Calabi–Yau submanifolds inside of a G 2 manifold. More specifically, we explain how to assign to a G 2 manifold (M, φ, Λ), with the calibration 3-form φ and an oriented 2-plane field Λ, a pair of parametrized tangent bundle valued 2- and 3-forms of M. These forms can then be used to define different complex and symplectic structures on certain 6-dimensional subbundles of T(M). When these bundles are integrated they give mirror CY manifolds. In a similar way, one can define mirror dual G 2 manifolds inside of a Spin(7) manifold (N 8, Ψ). In case N 8 admits an oriented 3-plane field, by iterating this process we obtain Calabi–Yau submanifold pairs in N whose complex and symplectic structures determine each other via the calibration form of the ambient G 2 (or Spin(7)) manifold.
06 May 2010
TL;DR: In this article, the authors studied the singularities of geodesic flows on surfaces with nonisolated singular points that form a smooth curve (like a cuspidal edge) and the normal forms of the corresponding direction field on the tangent bundle of the plane of local coordinates.
Abstract: This paper is a study of singularities of geodesic flows on surfaces with nonisolated singular points that form a smooth curve (like a cuspidal edge). The main results of the paper are normal forms of the corresponding direction field on the tangent bundle of the plane of local coordinates and the projection of its trajectories to the surface.
TL;DR: In this article, the warped Sasaki-Matsumoto metric *G is introduced for the warped product Finsler manifold, and it is shown if the warped function f is not a constant, then *G on TM∘ is bundlelike for the warp vertical foliation V*(TM∘) if and only if F1n1=(M1,F1) and F2n2=(M2,F2) are Riemannian manifolds.
Abstract: Let Fn=(M,F) be a Finsler manifold and G be the Sasaki–Matsumoto metric on TM∘. Bejancu and Farran [“Finsler geometry and natural foliations on the tangent bundle,” Rep. Math. Phys. 58, 131 (2006)] proved that Fn=(M,F) is a Riemannian manifold if and only if the Sasaki–Matsumoto metric G on TM∘ is bundlelike for the vertical foliation. Let Fn1+n2=(M1×fM2,F) be the warped product Finsler manifold. In this paper the warped Sasaki–Matsumoto metric *G is introduced for the warped product Finsler manifold, and it is shown if the warped function f is not a constant, then *G on TM∘ is bundlelike for the warped vertical foliation V*(TM∘) if and only if F1n1=(M1,F1) and F2n2=(M2,F2) are Riemannian manifolds.
19 Mar 2010
TL;DR: In this article, it was shown that the tangent bundle of a simple linear algebraic group G/P is stable under the assumption that P is a parabolic subgroup of G.
Abstract: Let P be a parabolic subgroup of a complex simple linear algebraic groupG. We prove that the tangent bundle T (G/P) is stable.
Journal Article•
TL;DR: Boeckx and Vanhecke as mentioned in this paper showed that if arbitrary Riemannian g-natural metrics are considered, then the geodesic flow is still a harmonic vector field, and also defines a harmonic map under some conditions on the gnatural metrics.
Abstract: Let (M,g) be a Riemannian manifold. We equip the unit tangent sphere bundle T1 M of (M,g) and its unit tangent sphere bundle Tr T1M of radius r>0 with arbitrary Riemannian g-natural metrics. When (M,g) is two-point homogeneous and both T1 M and T1T1M are equipped with the Sasaki metrics, the geodesic flow vector field is harmonic and determines a harmonic map [E. Boeckx and L. Vanhecke, Harmonic and minimal vector fields on tangent and unit tangent bundles, Diff. Geom. Appl., 13 (2000), 77-93]. We prove that if arbitrary Riemannian g-natural metrics are considered, then the geodesic flow is still a harmonic vector field, and it also defines a harmonic map under some conditions on the g-natural metrics. This permits to exhibit large families of harmonic maps defined in a compact Riemannian manifold and having a target space with a highly nontrivial geometry. In particular, explicit examples are provided on the unit tangent sphere bundle of the sphere S n and the flat torus Tn. Moreover, the geodesic flow being a Killing vector field is characterized in terms of harmonicity of the corresponding map and of properties of the base manifold.
TL;DR: In this paper, the authors improved Tanno's result that a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, and thus it is an automorphism of M by waiving the "strictness" in the hypothesis.
Abstract: First we improve a result of Tanno that says "If a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, then it is an infinitesimal automorphism of M" by waiving the "strictness" in the hypothesis. Next, we prove that a (k, μ)-contact manifold admitting a non-Killing conformal vector field is either Sasakian or has k = –n – 1, μ = 1 in dimension > 3; and Sasakian or flat in dimension 3. In particular, we show that (i) among all compact simply connected (k, μ)-contact manifolds of dimension > 3, only the unit sphere S2n+1 admits a non-Killing conformal vector field, and (ii) a conformal vector field on the unit tangent bundle of a space-form of dimension > 2 is necessarily Killing.
TL;DR: In this paper, it was shown that a compact contact threefold which is bimeromorphically equivalent to a Kaehler manifold and not rationally connected is the projectivised tangent bundle of a kaehler surface.
Abstract: We prove that a compact contact threefold which is bimeromorphically equivalent to a Kaehler manifold and not rationally connected is the projectivised tangent bundle of a Kaehler surface.
TL;DR: In this paper, the authors studied the distribution of geodesic arcs with respect to equilibrium states of the flow on a closed manifold, equipped with a $\mathcal{C}^{\infty}$ Riemannian metric.
Abstract: Using the works of Mane \cite{Ma} and Paternain \cite{Pat} we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a $\mathcal{C}^{\infty}$ Riemannian metric. We prove large deviations lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.
TL;DR: In this article, it was shown that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class.
Abstract: Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit vector field tangent to the fibres as a cross-section of the unit tangent bundle of the sphere, and prove that it is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Previous attempts to find a mathematical sense in which Hopf fibrations and Hopf vector fields are optimal have met with limited success.
TL;DR: In this paper, the authors give conditions under which the tangent bundle of a complete intersection on a Fano variety of Picard number one is stable or strongly stable over an algebraically closed field.
Abstract: Let X be a Fano variety of Picard number one defined over an algebraically closed field. We give conditions under which the tangent bundle of a complete intersection on X is stable or strongly stable.
TL;DR: In this paper, the authors investigate horizontal conformality of a differential of a map between Riemannian manifolds, where the tangent bundles are equipped with Cheeger-Gromoll-type metrics.
Abstract: We investigate horizontal conformality of a differential of a map between Riemannian manifolds, where the tangent bundles are equipped with Cheeger–Gromoll-type metrics. As a corollary, we characterize the differential of a map as a harmonic morphism.
10 Aug 2010
TL;DR: In this article, the curvature of the pull back of a fiber bundle from a C∞ principal G-bundle to a C ∞ manifold has been investigated, where the fiber bundle has a tautological connection.
Abstract: Let M be a C∞ manifold and G a Lie a group. Let EG be a C∞ principal G-bundle over M. There is a fiber bundle C(EG) over M whose smooth sections correspond to the connections on EG. The pull back of EG to C(EG) has a tautological connection. We investigate the curvature of this tautological connection.
Posted Content•
TL;DR: In this paper, rank-one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesic flows are constructed, such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodeic flow.
Abstract: We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. Finally, we give examples of complete rank one surfaces for which the non wandering set of the geodesic flow is connected, the periodic orbits are dense in that set, yet the geodesic flow is not transitive in restriction to its non wandering set.
Posted Content•
TL;DR: In this paper, the authors give an exposition on the Holmes-Thompson theory developed by Alvarez and show that it can also be obtained from the symplectic structure on the tangent bundle of the Riemannian manifold, the manifold of the Minkowski unit sphere.
Abstract: In this article, we give an exposition on the Holmes-Thompson theory developed by Alvarez. The space of geodesics in Minkowski space has a symplectic structure which is induced by the projection from the sphere- bundle. we show that it can be also obtained from the symplectic structure on the tangent bundle of the Riemannian manifold, the tangent bundle of the Minkowski unit sphere. We give detailed descriptions and expositions on Holmes-Thompson volumes in Minkowski space by the symplectic structure and the Crofton measures for them. For the Minkowski plane, a normed two dimensional space, we express the area explicitly in an integral geometry way, by putting a measure on the plane, which gives an extension of Alvarez's result for higher dimensional cases. 1. Introductions
01 Jan 2010
TL;DR: In this article, the authors considered a Finsler vector bundle and showed that the indicatrix is a totally umbilical submanifold in Ex of constant mean curvature 1.
Abstract: U Abstract. We consider a Finsler vector bundle i.e. a vector bundle : (E;p;M) endowed with a smooth function F : E ! IR; (x;y)7! F (x;y) that is positively homogeneous of degree 1 with respect to the variables y in bres of . Then F (x;y) = 1 with a xed x denes the indicatrix of the given Finsler bundle in the bre Ex and F (x;y) = 1 for everyx andy is its indicatrix bundle. We show in Section 2 that the indicatrix is a totally umbilical submanifold in Ex of constant mean curvature 1. The indicatrix bundle is a submanifold of En 0 . As- suming that is endowed with a nonlinear connection compatible with F and the base M is a Riemannian manifold we dene a Riemannian metric on En 0 and determine the normal to the indicatrix bundle.
TL;DR: In this paper, it was shown that the Hopf field is stable on S n ( r ) / Γ, Γ ≠ { Id }, unless Γ is trivial.
Abstract: The volume of a unit vector field V of a Riemannian manifold ( M , g ) is the volume of its image V ( M ) in the unit tangent bundle endowed with the Sasaki metric. Unit Hopf vector fields, that is, unit vector fields that are tangent to the fiber of a Hopf fibration S n → C P n − 1 2 (n odd) are well known to be critical for the volume functional on the round n-dimensional sphere S n ( r ) for every radius r > 1 . Regarding the Hessian, it turns out that its positivity actually depends on the radius. Indeed, in Borrelli and Gil-Medrano (2006) [2], it is proven that for n ⩾ 5 there is a critical radius r c = 1 n − 4 such that Hopf vector fields are stable if and only if r ⩽ r c . In this paper we consider the question of the existence of a critical radius for space forms M n ( c ) (n odd) of positive curvature c. These space forms are isometric quotients S n ( r ) / Γ of round spheres and naturally carry a unit Hopf vector field which is critical for the volume functional. We prove that r c = + ∞ , unless Γ is trivial. So, in contrast with the situation for the sphere, the Hopf field is stable on S n ( r ) / Γ , Γ ≠ { Id } , whatever the radius.
Posted Content•
TL;DR: In this paper, the following theorem is proved: if M is an n-dimensional (n>2) submanifold of a Riemannian manifold N, and through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p, then M is totally geodesic in N.
Abstract: The following Theorem is proved: Let M be an n-dimensional (n>2) submanifold of a Riemannian manifold N. Suppose that through each point p of M there exist two (n-1)-dimensional extrinsic spheres of N, which are contained in M in a neighbourhood of p and are tangent to each other at p. Then M is totally geodesic in N or an extrinsic sphere of N.
TL;DR: In this article, a manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action, where the geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geode spray.
Abstract: A manifold with an arbitrary affine connection is considered and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dy- namics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation.
TL;DR: In this article, the authors generalized the projection results concerning the dimension spectrum of projected measures on R n to parametrized families of transversal mappings between smooth manifolds and measures on them.
Abstract: In this work we first generalize the projection results concerning the dimension spectrum of projected measures on R n to parametrized families of transversal mappings between smooth manifolds and measures on them. The projection theorems for the lower q-dimension were first considered in (FO) and (HK). Theorems for the upper q-dimension were first considered in (FO) and (JJ). After proving the generalized results, we compute for 1 < q • 2 the lower and the upper q-dimensions of the natural projection of a probability measure which is invariant under the geodesic flow on the unit tangent bundle of a two-dimensional Riemann manifold.