Showing papers on "Unit tangent bundle published in 2012"
TL;DR: In this article, an action integral on the unit tangent bundle is constructed for Finsler spacetimes, which are generalizations of Lorentzian metric manifolds and satisfy necessary causality properties.
Abstract: We construct gravitational dynamics for Finsler spacetimes in terms of an action integral on the unit tangent bundle. These spacetimes are generalizations of Lorentzian metric manifolds which satisfy necessary causality properties. A coupling procedure for matter fields to Finsler gravity completes our new theory that consistently becomes equivalent to Einstein gravity in the limit of metric geometry. We provide a precise geometric definition of observers and their measurements and show that the transformations, by means of which different observers communicate, form a groupoid that generalizes the usual Lorentz group. Moreover, we discuss the implementation of Finsler spacetime symmetries. We use our results to analyze a particular spacetime model that leads to Finsler geometric refinements of the linearized Schwarzschild solution.
139 citations
TL;DR: In this article, the authors study a model of geometry of vision due to Petitot, Citti, and Sarti, where the primary visual cortex V1 lifts an image from a corrupted image to the bundle of directions of the plane, and then the corrupted image is reconstructed by minimizing the energy necessary for activation of the orientation columns corresponding to regions in which the image is corrupted.
Abstract: In this paper we study a model of geometry of vision due to Petitot, Citti, and Sarti. One of the main features of this model is that the primary visual cortex V1 lifts an image from $\mathbb{R}^2$ to the bundle of directions of the plane. Neurons are grouped into orientation columns, each of them corresponding to a point of this bundle. In this model a corrupted image is reconstructed by minimizing the energy necessary for the activation of the orientation columns corresponding to regions in which the image is corrupted. The minimization process intrinsically defines a hypoelliptic heat equation on the bundle of directions of the plane. In the original model, directions are considered both with and without orientation, giving rise, respectively, to a problem on the group of rototranslations of the plane $SE(2)$ or on the projective tangent bundle of the plane $PT\mathbb{R}^2$. We provide a mathematical proof of several important facts for this model. We first prove that the model is mathematically consis...
61 citations
TL;DR: This work exploits the observation that curve completion is an early visual process to formalize the problem in the unit tangent bundle R2 × S1, which abstracts the primary visual cortex (V1) and facilitates exploration of basic principles from which perceptual properties are later derived rather than imposed.
Abstract: Visual curve completion is a fundamental perceptual mechanism that completes the missing parts (e.g., due to occlusion) between observed contour fragments. Previous research into the shape of completed curves has generally followed an “axiomatic” approach, where desired perceptual/geometrical properties are first defined as axioms, followed by mathematical investigation into curves that satisfy them. However, determining psychophysically such desired properties is difficult and researchers still debate what they should be in the first place. Instead, here we exploit the observation that curve completion is an early visual process to formalize the problem in the unit tangent bundle R2 × S1, which abstracts the primary visual cortex (V1) and facilitates exploration of basic principles from which perceptual properties are later derived rather than imposed. Exploring here the elementary principle of least action in V1, we show how the problem becomes one of finding minimum-length admissible curves in R2 × S1. We formalize the problem in variational terms, we analyze it theoretically, and we formulate practical algorithms for the reconstruction of these completed curves. We then explore their induced visual properties vis-a-vis popular perceptual axioms and show how our theory predicts many perceptual properties reported in the corresponding perceptual literature. Finally, we demonstrate a variety of curve completions and report comparisons to psychophysical data and other completion models.
54 citations
TL;DR: In this paper, the authors consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate for mixing.
Abstract: We consider smooth time-changes of the classical horocycle flows on
the unit tangent bundle of a compact hyperbolic surface and prove
sharp bounds on the rate of equidistribution and the rate of
mixing. We then derive results on the spectrum of smooth time-changes
and show that the spectrum is absolutely continuous with respect to
the Lebesgue measure on the real line and that the maximal spectral
type is equivalent to Lebesgue.
46 citations
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TL;DR: This work proposes an amplification of the ML, called Tangent Bundle ML, in which the proximity not only between the original manifold and its estimator but also between their tangent spaces is required.
Abstract: One of the ultimate goals of Manifold Learning (ML) is to reconstruct an unknown nonlinear low-dimensional manifold embedded in a high-dimensional observation space by a given set of data points from the manifold. We derive a local lower bound for the maximum reconstruction error in a small neighborhood of an arbitrary point. The lower bound is defined in terms of the distance between tangent spaces to the original manifold and the estimated manifold at the considered point and reconstructed point, respectively. We propose an amplification of the ML, called Tangent Bundle ML, in which the proximity not only between the original manifold and its estimator but also between their tangent spaces is required. We present a new algorithm that solves this problem and gives a new solution for the ML also.
32 citations
TL;DR: In this paper, the geometric properties of the manifold of states described as (uniform) matrix product states are studied, and the main interest is in the states living in the tangent space to the base manifold, which have been shown to be interesting in relation to time dependence and elementary excitations.
Abstract: We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e. the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a K\"ahler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.
20 citations
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TL;DR: In this article, the authors extend the notion of a Parseval frame for a fixed Hilbert space to that of a movingParseval Frame for a vector bundle over a manifold, and prove that a sequence of vector fields is a moving Parsevel Frame for the tangent bundle of a Riemannian manifold if and only if the vector fields are the orthogonal projection of the moving orthonormal basis for a larger vector bundle.
Abstract: Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for a vector bundle over a manifold. Many vector bundles do not have a moving basis, but in contrast to this every vector bundle over a paracompact manifold has a moving Parseval frame. We prove that a sequence of sections of a vector bundle is a moving Parseval frame if and only if the sections are the orthogonal projection of a moving orthonormal basis for a larger vector bundle. In the case that our vector bundle is the tangent bundle of a Riemannian manifold, we prove that a sequence of vector fields is a Parseval frame for the tangent bundle of a Riemannian manifold if and only if the vector fields are the orthogonal projection of a moving orthonormal basis for the tangent bundle of a larger Riemannian manifold.
18 citations
TL;DR: In this article, Oh and Shah showed that the skinning measures, when finite, of the equidistant hypersurfaces to C equidistribute to the Bowen-Margulis measure on the unit tangent bundle of M, under additional assumptions on the rate of mixing.
Abstract: Let C be a locally convex subset of a negatively curved Riemannian manifold M. We define the skinning measure on the outer unit normal bundle to C in M by pulling back the Patterson-Sullivan measures at infinity, and give a finiteness result of skinning measures, generalising the work of Oh and Shah, with different methods. We prove that the skinning measures, when finite, of the equidistant hypersurfaces to C equidistribute to the Bowen-Margulis measure on the unit tangent bundle of M, assuming only that the Bowen-Margulis measure is finite and mixing for the geodesic flow. Under additional assumptions on the rate of mixing, we give a control on the rate of equidistribution.
18 citations
TL;DR: The main purpose of as mentioned in this paper is to investigate the paraholomorphy property of the Sasaki and Cheeger-Gromoll metrics by using compatible paracomplex stuctures on the tangent bundle.
Abstract: The main purpose of this article is to investigate the paraholomorphy property of the Sasaki and Cheeger–Gromoll metrics by using compatible paracomplex stuctures on the tangent bundle.
14 citations
TL;DR: In this paper, it was shown that if the bottom of the spectrum of a flow is point-wise pinched and is integrable, then the flow has regular distortion along unstable manifolds over the tangent bundle.
Abstract: In this paper we study a certain regularity property of
$C^2$ Axiom A flows $\phi_t$ over basic sets $Λ$ related to diameters of balls in Bowen's
metric, which we call regular distortion along unstable manifolds. The motivation to
investigate the latter comes from the study of spectral properties of Ruelle transfer
operators in [21]. We prove that if the bottom of the spectrum of $d\phi_t$ over $E^u_{|Λ}$ is point-wisely pinched and
integrable, then the flow has regular distortion along unstable manifolds over $Λ$.
In the process, under the same conditions, we show that locally the flow is Lipschitz conjugate to
its linearization over the `pinched part' of the unstable tangent bundle.
14 citations
TL;DR: In this article, the authors studied the paraholomorphy property of two Riemannian metrics of Cheeger Gromoll type depending on one parameter and two parameters by using compatible paracomplex structures J a and J a, b on the tangent bundle T M.
TL;DR: In this paper, it was shown that there is no mathematical equivalence with Einstein's vacuum field equations in space-times of 2D dimensions, with two times, after a d+d Kaluza-Klein-like decomposition of the 2d-dim scalar curvature R is performed and involving the introduction of a nonlinear connection.
Abstract: The generalized (vacuum) field equations corresponding to gravity on curved 2d-dimensional (dim) tangent bundle/phase spaces and associated with the geometry of the (co)tangent bundle TMd-1, 1(T*Md-1, 1) of a d-dim space–time Md-1, 1 are investigated following the strict distinguished d-connection formalism of Lagrange–Finsler and Hamilton–Cartan geometry. It is found that there is no mathematical equivalence with Einstein's vacuum field equations in space–times of 2d dimensions, with two times, after a d+d Kaluza–Klein-like decomposition of the 2d-dim scalar curvature R is performed and involving the introduction of a nonlinear connection . The physical applications of the 4-dim phase space metric solutions found in this work, corresponding to the cotangent space of a 2-dim space–time, deserve further investigation. The physics of two times may be relevant in the solution to the problem of time in quantum gravity and in the explanation of dark matter. Finding nontrivial solutions of the generalized gravitational field equations corresponding to the 8-dim cotangent bundle (phase space) of the 4-dim space–time remains a challenging task.
TL;DR: In this paper, the authors complete the classification of compact inner symmetric spaces with weakly complex tangent bundles by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic.
Abstract: We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.
TL;DR: The Program for New Century Excellent Talents in Fujian Province of China as discussed by the authors, which is a part of the National Natural Science Foundation of China (NFF), has been used to train Chinese students.
Abstract: Program for New Century Excellent Talents in Fujian Province; Natural Science Foundation of China [10971170, 10601040]
TL;DR: In this article, the authors studied the natural contact structure on the tangent bundle and tangent sphere bundles of a Riemannian manifold M with radius function r and deduced the equations of induced metric connections on those bundles.
Abstract: Natural metric structures on the tangent bundle and tangent sphere bundles SrM of a Riemannian manifold M with radius function r enclose many important unsolved problems. Admitting metric connections on M with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of TM to the almost Hermitian category. Our purpose is the study of the natural contact structure on SrM and the G2-twistor space of any oriented Riemannian 4-manifold.
TL;DR: In this paper, a generalised spin structure on a 2-dimensional hyperbolic orbifold is investigated and conditions on the existence of such structures are given, and the moduli space of taut contact circles on left-quotients of the 3-dimensional geometry is described.
Abstract: Generalised spin structures, or $r$-spin structures, on a $2$-dimensional orbifold $\Sigma$ are $r$-fold fibrewise connected coverings (also called $r$\textsuperscript{th} roots) of its unit tangent bundle $ST\Sigma$. We investigate such structures on hyperbolic orbifolds. The conditions on $r$ for such structures to exist are given. The action of the diffeomorphism group of $\Sigma$ on the set of $r$-spin structures is described, and we determine the number of orbits under this action and their size. These results are then applied to describe the moduli space of taut contact circles on left-quotients of the $3$-dimensional geometry $\widetilde{\mathrm{SL}}_{2}$.
TL;DR: In this paper, the authors give conditions under which the tangent bundle endowed with such a structure and with a general natural lifted metric is a Riemannian almost product (locally product) or an (almost) para-Hermitian manifold.
Abstract: We find the almost product (locally product) structures of general natural lift type on the tangent bundle of a Riemannian manifold. We get the conditions under which the tangent bundle endowed with such a structure and with a general natural lifted metric is a Riemannian almost product (locally product) or an (almost) para-Hermitian manifold. We give a characterization of the general natural (almost) para-Hermitian structures, which are (almost) para-K a hlerian on the tangent bundle.
TL;DR: In this paper, the Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundles and as one-form the Cartan form.
Abstract: The aim of this paper is to study from the point of view of linear connections the data with M a smooth (n+p)-dimensional real manifold, an n-dimensional manifold semi-Riemannian distribution on M, the conformal structure generated by g and W a Weyl substructure: a map such that W(ḡ) = W(g) - du, ḡ = eug;u ∈ C∞(M). Compatible linear connections are introduced as a natural extension of similar notions from Weyl geometry and such a connection is unique if a symmetry condition is imposed. In the foliated case the local expression of this unique connection is obtained. The notion of Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundle projection and as one-form the Cartan form.
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TL;DR: In this paper, the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three was studied.
Abstract: Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space FTM on the bundle of frames in the tangent bundle of M. We show that FTM is the group completion of LM, and prove that it is obtained by localizing LM with respect to adding a "small twist".
TL;DR: In this paper, the authors extend the notion of tangent bundle to a $z$ graded bundle, which has a Lie algebroid structure and they can develop notions semi-riemannian metrics, Levi-civita connection, and curvature on it.
Abstract: In this paper we will extend the notion of tangent bundle to a $\z$ graded tangent bundle. This graded bundle has a Lie algebroid structure and we can develop notions semi-riemannian metrics, Levi-civita connection, and curvature, on it. In case of space-times manifolds, even part of the tangent bundle is related to space and time structures(gravity) and odd part is related to mass distribution in space-time. In this structure, mass becomes part of the geometry, and Einstein field equation can be reconstructed in a new simpler form. The new field equation is purely geometric.
09 Feb 2012
TL;DR: In this paper, a higher-order tangent connection is defined on a fibered manifold by assuming the structure of a higher order tangent bundle and using White's sector-forms on these bundles.
Abstract: To define a higher order connection on a fibered manifold one can use the sections of nonholonomic jet prolongations. However, a more natural approach seems to be the one assuming the structure of a higher-order tangent bundle and using White's sector-forms on these bundles.
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TL;DR: In this paper, the splitting of the normal bundle of rational curves was studied with apolarity theory and some particular subvarieties in some Hilbert scheme of rational curve, defined by splitting type of normal bundle and the restricted tangent bundle.
Abstract: As in our previous work [1] we address the problem to determine the splitting of the normal bundle of rational curves. With apolarity theory we are able to characterize some particular subvarieties in some Hilbert scheme of rational curves, defined by the splitting type of the normal bundle and the restricted tangent bundle.
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TL;DR: In this article, an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume was shown.
Abstract: The main result of this paper is an effective count for Apollonian circle packings that are either bounded or contain two parallel lines. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3-manifold of infinite volume, whose fundamental group has critical exponent bigger than 1. We also discuss applications to Affine sieves. Analogous results for surfaces are treated as well.
TL;DR: In this article, it was shown that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable.
Abstract: We prove that a pair (F = vector sub-bundle of TM, its annihilator) yields an almost Dirac structure which is Dirac if and only if F is Lie integrable. Then a flat Ehresmann connection on a fiber bundle ξ yields two complementary, but not orthogonally, Dirac structures on the total space M of ξ. These Dirac structures are also Lagrangian sub-bundles with respect to the natural almost symplectic structure of the big tangent bundle of M. The tangent bundle in Riemannian geometry is discussed as particular case and the 3-dimensional Heisenberg space is illustrated as example. More generally, we study the Bianchi–Cartan–Vranceanu metrics and their Hopf bundles.
01 Jan 2012
TL;DR: In this paper, the authors completely classify Riemannian $g$-natural metrics of constant sectional curvature on the unit tangent sphere bundle $T_1 M$ of a Riemanian manifold.
Abstract: We completely classify Riemannian $g$-natural metrics of constant sectional curvature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$. Since the base manifold $M$ turns out to be necessarily two-dimensional, weaker curvature conditions are also investigated for a Riemannian $g$-natural metric on the unit tangent sphere bundle of a Riemannian surface.
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TL;DR: In this article, the authors consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and rate of mixing.
Abstract: We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the spectrum of smooth time-changes and show that the spectrum is absolutely continuous with respect to the Lebesgue measure on the real line and that the maximal spectral type is equivalent to Lebesgue.
TL;DR: In this paper, the authors introduced a class of g-natural metrics Ga,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki-Matsumoto metric and Miron metric.
Abstract: Recently the third author studied horizontal Laplacians in real Finsler vector bundles and complex Finsler manifolds. In this paper, we introduce a class of g-natural metrics Ga,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki–Matsumoto metric and Miron metric. We obtain the Weitzenbock formula of the horizontal Laplacian associated to Ga,b, which is a second-order differential operator for general forms on tangent bundle. Using the horizontal Laplacian associated to Ga,b, we give some characterizations of certain objects which are geometric interest (e.g. scalar and vector fields which are horizontal covariant constant) on the tangent bundle. Furthermore, Killing vector fields associated to Ga,b are investigated.
Journal Article•
TL;DR: In this article, it was shown that the fourfold Whitney sum of the tangent bundle of real projective planes of any three dimensional nontrivial real G-representation is equivariantly a product bundle.
Abstract: let G be a nontrivial cyclic group of odd order. In the present paper, we will prove that the fourfold Whitney sum of the tangent bundle of real projective plane of any three dimensional nontrivial real G-representation is equivariantly a product bundle.
TL;DR: In this article, the authors investigated conformality of a mapping between Riemannian manifolds if the tangent bundles are equipped with a generalized metric of Cheeger-Gromoll type.
Abstract: We investigate conformality of the differential of a mapping between Riemannian manifolds if the tangent bundles are equipped with a generalized metric of Cheeger-Gromoll type.
TL;DR: In this article, the Ricci curvature on the unit tangent bundle of a complete Finsler manifold M without conjugate points is shown to be non-positive and vanishes only if M is flat.
Abstract: We prove that the integral of the Ricci curvature on the unit tangent bundle SM of a complete Finsler manifold M without conjugate points is nonpositive and vanishes only if M is flat, provided that the Ricci curvature on SM has an integrable positive or negative part.