Showing papers on "Unit tangent bundle published in 2014"
TL;DR: For Ricci-flat manifold with Euclidean volume growth, it was shown in this article that local tangent cones are unique if one tangent cone has a smooth cross-section.
Abstract: We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section.
60 citations
TL;DR: In this article, a maximal acceleration invariant quantum field is formulated in terms of the differential geometric structure of the spacetime tangent bundle, and the field is shown to have a physically based Planck-scale effective regularization and a spectral cutoff at the Planck mass.
Abstract: Maximal-acceleration invariant quantum fields are formulated in terms of the differential geometric structure of the spacetime tangent bundle. The simple special case is considered of a flat Minkowski space-time for which the bundle is also flat. The field is shown to have a physically based Planck-scale effective regularization and a spectral cutoff at the Planck mass.
27 citations
TL;DR: In this paper, it was shown that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds with respect to the Picard number.
Abstract: We prove that smooth Fano 5-folds with nef tangent bundles and Picard numbers greater than one are rational homogeneous manifolds.
23 citations
TL;DR: The index of symmetry as discussed by the authors measures how far a Riemannian manifold from being a symmetric space is from being one of the group types, and it is defined as a measure of how close a manifold is to being a group type.
Abstract: We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.
20 citations
TL;DR: In this article, a natural self-adjoint operator was defined for geodesic flows preserving a smooth measure on a closed manifold, which maps into the space of invariant distributions in the unit tangent bundle.
Abstract: For Anosov flows preserving a smooth measure on a closed manifold $\mathcal{M}$, we define a natural self-adjoint operator $\Pi$ which maps into the space of invariant distributions in $\cap_{u 0} H^{s}(\mathcal{M})$. We describe relations to Livsic theorem and recover regularity properties of cohomological equations using this operator. For Anosov geodesic flows on the unit tangent bundle $\mathcal{M}=SM$ of a compact manifold, we apply this theory to study questions related to $X$-ray transform on symmetric tensors on $M$: in particular we prove that injectivity implies surjectivity of X-ray transform, and we show injectivity for surfaces.
18 citations
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TL;DR: In this paper, it was shown that the pluriclosed flow preserves generalized Kahler structures with the extra condition $[J_+,J_-] = 0, a condition referred to as split tangent bundle.
Abstract: We show that the pluriclosed flow preserves generalized Kahler structures with the extra condition $[J_+,J_-] = 0$, a condition referred to as "split tangent bundle." Moreover, we show that in this in this case the flow reduces to a nonconvex fully nonlinear parabolic flow of a scalar potential function. We prove a number of a priori estimates for this equation, including a general estimate in dimension $n=2$ of Evans-Krylov type requiring a new argument due to the nonconvexity of the equation. The main result is a long time existence theorem for the flow in dimension $n=2$, covering most cases. We also show that the pluriclosed flow represents the parabolic analogue to an elliptic problem which is a very natural generalization of the Calabi conjecture to the setting of generalized Kahler geometry with split tangent bundle.
17 citations
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TL;DR: In this paper, the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds were studied and it was shown that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold.
Abstract: We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by trajectories not bounding a flat strip. This is done by showing that Dirac measures on periodic orbits are dense in that set. In the case of a compact surface, we get the following sharp result: ergod- icity is a generic property in the space of all invariant measures defined on the unit tangent bundle of the surface if and only if there are no flat strips in the universal cover of the surface. Finally, we show under suitable assumptions that generically, the invariant probability measures have zero entropy and are not strongly mixing.
16 citations
TL;DR: In this paper, it was shown that the flat product metric on a Riemannian manifold with boundary is scattering rigid, where the scattering data (loosely speaking) of the manifold is map $S:U+\partial M to U^-partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundaries that point outwards.
Abstract: We prove that the flat product metric on $D^n\times S^1$ is scattering rigid where $D^n$ is the unit ball in $\R^n$ and $n\geq 2$. The scattering data (loosely speaking) of a Riemannian manifold with boundary is map $S:U^+\partial M\to U^-\partial M$ from unit vectors $V$ at the boundary that point inward to unit vectors at the boundary that point outwards. The map (where defined) takes $V$ to $\gamma'_V(T_0)$ where $\gamma_V$ is the unit speed geodesic determined by $V$ and $T_0$ is the first positive value of $t$ (when it exists) such that $\gamma_V(t)$ again lies in the boundary. We show that any other Riemannian manifold $(M,\partial M,g)$ with boundary $\partial M$ isometric to $\partial(D^n\times S^1)$ and with the same scattering data must be isometric to $D^n\times S^1$. This is the first scattering rigidity result for a manifold that has a trapped geodesic. The main issue is to show that the unit vectors tangent to trapped geodesics in $(M,\partial M,g)$ have measure 0 in the unit tangent bundle.
16 citations
TL;DR: In this article, symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) flows on compact smooth three-dimensional manifolds were constructed.
Abstract: We construct symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+\epsilon}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^\infty$ surface has at least const $\times(e^{hT}/T)$ simple closed orbits of period less than $T$, whenever the topological entropy $h$ is positive -- and without further assumptions on the curvature.
12 citations
TL;DR: The vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler-Einstein manifold of classical type was shown in this article.
Abstract: We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.
12 citations
01 Jun 2014
TL;DR: In this article, the authors study lifts of hypersurfaces with quarter-symmetric semi-metric connection to tangent bundles and obtain certain results on totally geodesic and totally umbilical.
Abstract: The taking into considering lifting theory, we study lifts of hypersurfaces with Quarter -symmetric semi-metric connection to tangent bundles and we obtain certain results on totally geodesic and totally umbilical.
TL;DR: In this article, the authors extend the adiabatic limit formula for �-invariants by Bismut-Cheeger and Dai to Seifert fibrations and derive diffeomor-phism type of a new manifold of positive sectional curvature.
Abstract: We extend the adiabatic limit formula for �-invariants by Bismut-Cheeger and Dai to Seifert fibrations. Our formula contains a new contribution from the singular fibres that takes the form of a generalised Dedekind sum. As an application, we compute the Eells-Kuiper and t-invariants of cer- tain cohomogeneity one manifolds that were studied by Dearricott, Grove, Verdiani, Wilking, and Ziller. In particular, we determine the diffeomor- phism type of a new manifold of positive sectional curvature. Manifolds of positive sectional curvature are a rare phenomenon, and the differential topological conditions for the existence of positive sectional curva- ture metrics are not yet fully understood. For this reason, one is still inter- ested in finding new examples of positive sectional curvature metrics. Most known examples are quotients or biquotients of compact Lie groups. Coho- mogeneity one manifolds constitute another potential source of examples. By work of Grove, Wilking and Ziller (19), there are only two families (Pk), (Qk) of seven-dimensional cohomogeneity one manifolds, which possibly allow met- rics of positive sectional curvature and contain new examples. The space R mentioned there does not admit a positive sectional curvature metric by (29). Grove, Verdiani and Ziller have succeeded in (18) to construct a positive sec- tional curvature metric on P2, the first nontrivial member of the family (Pk). This manifold is homeomorphic to the unit tangent bundle T 1 S 4 of the four- dimensional sphere. In this paper, we will specify among other things an exotic spheresuch that P2 is diffeomorphic to the connected sum of T 1 S 4 and �. The manifolds Pk are highly connected with a finite cyclic cohomology group H 4 (Pk) ∼ π3(Pk) ∼ Z/kZ. By Crowley's work (7), it suffices to compute the Eells-Kuiper invariant µ(Pk) and a certain quadratic form q on H 4 (Pk) to determine their diffeomorphism types. These two invariants are classically defined on oriented spin manifolds N bounding Pk, but it is not clear how to construct such a manifold N directly. On the other hands, by results of Don- nelly (12), Kreck and Stolz (25) and Crowley and the author (9), both invariants can equivalently be expressed as linear combinations of η-invariants of certain Dirac operators and Cheeger-Chern-Simons correction terms on Pk itself. Hav- ing computed these invariants, one can write the spaces Pk as connected sums of exotic spheres and S 3 -bundles over S 4 using the computations for these bundles
TL;DR: In this article, a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundles of a manifold were constructed and used to obtain Finslerian geodesic coordinates.
Abstract: We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinates known from metric geometry to Finsler geometric manifolds and it turns out that they are identical to the Douglas-Thomas normal coordinates introduced earlier. We expand the fundamental geometry function of a Finsler spacetime in these new coordinates and find that it is constant to quadratic order. The quadratic order term comes with the non-linear curvature of the manifold. From physics these coordinates may be interpretation as the realisation of an Einstein elevator in Finslerian spacetime geometries.
TL;DR: In this paper, 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.
TL;DR: In this article, a new Courant algebroid on Poisson manifolds is studied, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged.
Abstract: We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its symmetry is a semidirect product of $\beta$-diffeomorphisms and $\beta$-transformations. It is a starting point of an alternative version of the generalized geometry based on the cotangent bundle, such as Dirac structures and generalized Riemannian structures. In particular, $R$-fluxes are formulated as a twisting of this Courant algebroid by a local $\beta$-transformations, in the same way as $H$-fluxes are the twist of the generalized tangent bundle. It is a $3$-vector classified by Poisson $3$-cohomology and it appears in a twisted bracket and in an exact sequence.
TL;DR: In this paper, the Lie algebroid generalized tangent bundle of a Kaluza-Klein vector bundle is introduced and the theory of general distinguished linear connections for this space is developed.
Abstract: Introducing the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza–Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza–Klein G-spaces and we develop the Einstein equations in this general framework.
TL;DR: In this article, the authors studied the geodesic flow on the unit tangent bundle of a rank one manifold and gave conditions under which all classical definitions of pressure of a Holder continuous potential coincide.
Abstract: We study the geodesic flow on the unit tangent bundle of a rank one manifold and we give conditions under which all classical definitions of pressure of a Holder continuous potential coincide. We provide a large deviation statement, which allows one to neglect (periodic) orbits that lack sufficient hyperbolic behaviour. Our results involve conditions on the potential, which take into consideration its properties in the non-hyperbolic part of the manifold. We draw some conclusions for the construction of equilibrium states.
TL;DR: In this article, the authors give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle over a Riemannian manifold when $E$ is endowed with a metric connection.
Abstract: We give a general description of the construction of weighted spherically symmetric metrics on vector bundle manifolds, i.e. the total space of a vector bundle $E\rightarrow M$, over a Riemannian manifold $M$, when $E$ is endowed with a metric connection. The tangent bundle of $E$ admits a canonical decomposition and thus it is possible to define an interesting class of two-weights metrics with the weight functions depending on the fibre norm of $E$; hence the generalized concept of spherically symmetric metrics. We study its main properties and curvature equations. Finally we focus on a few applications and compute the holonomy of Bryant-Salamon type $\mathrm{G}_2$ manifolds.
Journal Article•
TL;DR: In this article, the authors studied the topology of the space A(M) of (not necessarily contact) Anosov vector fields on the unit tangent bundle M of closed oriented hyperbolic surfaces Σ.
Abstract: The first half of this paper concerns the topology of the space A(M) of (not necessarily contact) Anosov vector fields on the unit tangent bundle M of closed oriented hyperbolic surfaces Σ. We show that there are countably infinite connected components of A(M), each of which is not simply connected. In the second part, we study contact Anosov flows. We show in particular that the time changes of contact Anosov flows forma C 1 -open subset of the space of the Anosov flows which leave a particular C ∞ volume form invariant, if the ambiant manifold is a rational homology sphere.
TL;DR: In this article, the authors studied the properties of the tangent bundle with a deformed complete lift metric, and showed that the deformed lift metric can be used to study the manifold properties of tangent bundles.
Abstract: In this paper, our aim is to study some properties of the tangent bundle with a deformed complete lift metric.
TL;DR: The main purpose of as mentioned in this paper is to study deformed Riemannian extensions in the cotangent bundle, and the curvature properties of metric connections for deformed RCM extensions are also investigated.
Abstract: The main purpose of this paper is to study deformed Riemannian extensions in the cotangent bundle. The curvature properties of metric connections for deformed Riemannian extensions are also investigated.
TL;DR: In this article, it was shown that the tangent bundle of a smooth Banach manifold admits a vector bundle structure over the manifold if and only if the manifold is endowed with a linear connection.
Abstract: The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. For a Banach manifold $M$ and a natural number $k$ first we determine a smooth manifold structure on $T^kM$ which also offers a fiber bundle structure for $(\pi_k,T^kM,M)$. Then we introduce a particular lift of linear connections on $M$ to geometrize $T^kM$ as a vector bundle over $M$. More precisely based on this lifted nonlinear connection we prove that $T^kM$ admits a vector bundle structure over $M$ if and only if $M$ is endowed with a linear connection. As a consequence applying this vector bundle structure we lift Riemannian metrics and Lagrangians from $M$ to $T^kM$. Also, using the projective limit techniques, we declare a generalized Fr\'echet vector bundle structure for $T^\infty M$ over $M$.
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TL;DR: In this paper, the tangent bundle decomposition generated by a system of ODEs was generalized to the case of second order PDEs, where a closed 1-form on the manifold of independent variables was specified, together with a transverse local vector field.
Abstract: We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent variables, together with a transverse local vector field. The resulting decomposition provides several natural curvature operators. The harmonic map equation is examined, and in this case both the 1-form and the vector field arise naturally.
TL;DR: In this article, the authors define the notion of the $k$'th order differential (T^kg:T^kM\longrightarrow T^kN$) for a given differentiable map $g$ between manifolds $M$ and $N$.
Abstract: The tangent bundle $T^kM$ of order $k$, of a smooth Banach manifold $M$ consists of all equivalent classes of curves that agree up to their accelerations of order $k$. In the previous work of the author he proved that $T^kM$, $1\leq k\leq \infty$, admits a vector bundle structure on $M$ if and only if $M $ is endowed with a linear connection or equivalently a connection map on $T^kM$ is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the notion of the $k$'th order differential $T^kg:T^kM\longrightarrow T^kN$ for a given differentiable map $g$ between manifolds $M$ and $N$. As we shall see, $T^kg$ becomes a vector bundle morphism if the base manifolds are endowed with $g$-related connections. In particular, replacing a connection with a $g$-related one, where $g:M\longrightarrow M$ is a diffeomorphism, follows invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combination of connection maps and manifold of $C^r$ maps we offer three examples to support our theory and reveal its interaction with the known problems such as Sasaki lift of metrics.
TL;DR: In this article, the canonical form in the non-holonomic second-order frame bundle of an n-dimensional manifold M is introduced as a restriction of the canonical forms of the bundle.
Abstract: Using horizontal n-bases of the tangent bundle of the linear frame bundle of an n-dimensional manifold M, the canonical form in the non-holonomic second-order frame bundle of M is introduced as a restriction of the canonical form of the bundle . This construction generalizes the ones in the corresponding semi-holonomic and holonomic second-order frame bundles. We prove that the natural projection of the set of all non-holonomic second-order frames of M into defines a principal bundle structure.
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TL;DR: In this article, the authors give an upper bound for the volume of a periodic geodesic on a surface with respect to the unit tangent bundle, which is linear in the geometric length of the geodesics.
Abstract: A periodic geodesic on a surface has a natural lift to the unit tangent bundle; when the complement of this lift is hyperbolic, its volume typically grows as the geodesic gets longer We give an upper bound for this volume which is linear in the geometric length of the geodesic
TL;DR: In this article, a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric was constructed.
Abstract: We construct a template with two ribbons that describes the topology of all periodic orbits of the geodesic flow on the unit tangent bundle to any sphere with three cone points with hyperbolic metric. The construction relies on the existence of a particular coding with two letters for the geodesics on these orbifolds.
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TL;DR: In this paper, a family of Kahler metrics on the tangent disk bundle of any real space-form or any of its quotients by discrete groups of isometries is presented.
Abstract: We nd a family of Kahler metrics on the tangent disk bundle of any real space-form or any of its quotients by discrete groups of isometries. Both zero-section and bres embed as real Lagrangians and totally geodesic submanifolds. The metric is complete in the non-negative curvature case and non-complete in the negative curvature case.
01 Jan 2014
TL;DR: In this article, a generalized almost tangent geometry on the big tangent bundle T big M associated to an almost-tangent structure on M is studied with a special view towards integrability, and the compatibility with a class of generalized Riemannian metrics is discussed.
Abstract: A generalized almost tangent structure on the big tangent bundle T big M associated to an almost tangent structure on M is con- sidered and several features of it are studied with a special view towards integrability. Deformation under a fl- or a B-fleld transformation and the compatibility with a class of generalized Riemannian metrics are discussed. Also, a notion of tangentomorphism is introduced as a difieomorphism f preserving the (generalized) almost tangent geometry and some remarka- ble subspaces are proved to be invariant with respect to the lift of f.
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TL;DR: In this paper, the authors consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2torus, and show that ergodicity is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle.
Abstract: We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent bundle. On the other hand, using results of J. Franks and M. Handel, we prove that ergodicity and dense orbits cannot occur in the full unit tangent bundle of the 2-sphere, if the Finsler metric has positive flag curvatures and at least two closed geodesics. In the case of the 2-torus, we show that ergodicity is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle of the 2-torus.