Showing papers on "Unit tangent bundle published in 2021"

Local limit theorem in negative curvature

Abstract: Consider the heat kernel ℘(t,x,y) on the universal cover M˜ of a closed Riemannian manifold of negative sectional curvature. We show the local limit theorem for ℘: limt→∞t3∕2eλ0t℘(t,x,y)=C(x,y), where λ0 is the bottom of the spectrum of the geometric Laplacian and C(x,y) is a positive λ0-harmonic function which depends on x,y∈M˜. We also show that the λ0-Martin boundary of M˜ is equal to its topological boundary. The Martin decomposition of C(x,y) gives a family of measures {μxλ0} on ∂M˜. We show that {μ xλ0} is a family minimizing the energy or Mohsen’s Rayleigh quotient. We apply the uniform Harnack inequality on the boundary ∂M˜ and the uniform three-mixing of the geodesic flow on the unit tangent bundle SM for suitable Gibbs–Margulis measures.

Generic Dynamical Properties of Connections on Vector Bundles

Abstract: Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $ abla^{\mathcal{E}}$ on the vector bundle $\mathcal{E}$. First of all, we show that twisted Conformal Killing Tensors (CKTs) are generically trivial when $\dim(M) \geq 3$, answering an open question of Guillarmou-Paternain-Salo-Uhlmann. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $ abla^{\mathrm{End}(\mathcal{E})}$ on the endomorphism bundle $\mathrm{End}(\mathcal{E})$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e. the geodesic flow is Anosov on the unit tangent bundle), the connections are generically $\textit{opaque}$, namely there are no non-trivial subbundles of $\mathcal{E}$ which are generically preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called $\textit{operators of uniform divergence type}$, and on perturbative arguments from spectral theory (especially on the theory of Pollicott-Ruelle resonances in the Anosov case).

Patterson--Sullivan densities in convex projective geometry

Abstract: For any rank-one convex projective manifold with a compact convex core, we prove that there exists a unique probability measure of maximal entropy on the set of unit tangent vectors whose geodesic is contained in the convex core, and that it is mixing. We use this to establish asymptotics for the number of closed geodesics. In order to construct the measure of maximal entropy, we develop a theory of Patterson--Sullivan densities for general rank-one convex projective manifolds. In particular, we establish a Hopf--Tsuji--Sullivan--Roblin dichotomy, and prove that, when it is finite, the measure on the unit tangent bundle induced by a Patterson--Sullivan density is mixing under the action of the geodesic flow.

Surgery on Anosov flows using bi-contact geometry

Abstract: We describe a Dehn type surgery along a Legendrian-transverse knot K in a bi-contact structure. We show that, when the bi-contact structure defines an Anosov flow, there is a strong connection between the Anosovity of the new flow and contact geometry. We give an application to the geodesic flow on the unit tangent bundle of an hyperbolic surface. In contrast to the existing Dehn type surgeries on contact Anosov flows, for a vast class of knots our procedure does not require any restriction on the slope of the twist to generate new contact Anosov flows. We finally show that there are connections between our construction and the ones defined by Handel-Thurston, Fried-Goodman and Foulon-Hasselblatt.

Legendre curves and the singularities of ruled surfaces obtained by using rotation minimizing frame

Abstract: UDC 514.7 In this paper, Legendre curves in unit tangent bundle are given using rotation minimizing vector fields. Ruled surfaces corresponding to these curves are represented. Singularities of these ruled surfaces are also analyzed and classified.

Unique ergodicity of the horocyclic flow on nonpositively curved surfaces

Abstract: On the unit tangent bundle of a nonflat compact nonpositively curved surface, we prove that there is a unique probability Borel measure invariant by the horocyclic flow which gives full measure to the set of rank 1 vectors recurrent by the geodesic flow. If we assume in addition that the surface has no flat strips, we show that the horocyclic flow is uniquely ergodic. These results are valid for any parametrization of the horocyclic flow.

On $g$-natural conformal vector fields on unit tangent bundles

Abstract: We study conformal and Killing vector fields on the unit tangent bundle, over a Riemannian manifold, equipped with an arbitrary pseudo-Riemannian g-natural metric. We characterize the conformal and Killing conditions for classical lifts of vector fields and we give a full classification of conformal fiber-preserving vector fields on the unit tangent bundle endowed with an arbitrary pseudo-Riemannian Kaluza-Klein type metric.

On the biharmonicity of vector fields and unit vector fields

Abstract: Let $(M,g)$ be a compact Riemannian manifold. Equipping its tangent bundle $TM$ (resp. unit tangent bundle $T_1M$) by a pseudo-Riemannian $g$-natural metric $G$ (resp. $\tilde{G}$), we study the biharmonicty of vector fields (resp. unit vector fields) as maps $(M,g) \rightarrow (TM,G)$ (resp. $(M,g) \rightarrow (T_1M,\tilde{G})$) as well as critical points of the bienergy functional restricted to the set $\mathfrak{X}(M)$ (resp. $\mathfrak{X}^1(M)$) of vector fields (resp. unit tangent bundles) on $M$. Contrary to the Sasaki metric on $TM$, where the two notions are equivalent to the harmonicity of the vector field and then to its parallelism, we prove that for large classes of $g$-natural metrics on $TM$ the two notions are not equivalent. Furthermore, we give examples of vector fields which are biharmonic as critical points of the bienergy functional restricted to $\mathfrak{X}(M)$, but are not biharmonic maps. We provide equally examples of proper biharmonic vector fields (resp. unit vector fields), i.e. those which are biharmonic without being harmonic.

A connecting theorem for geodesic flows on the 2-torus

Abstract: We use a result of J. Mather on the existence of connecting orbits for compositions of monotone twist maps of the cylinder to prove the existence of connecting geodesics on the unit tangent bundle $ST^2$ of the 2-torus in regions without invariant tori.

Distribution in the unit tangent bundle of the geodesics of given type

Abstract: Recall that two geodesics in a negatively curved surface $S$ are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure $\mathfrak{m}^S$ on $T^1S$. We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.

Vortices over Riemann surfaces and dominated splittings

Abstract: We associate a flow $\phi$ to a solution of the vortex equations on a closed oriented Riemannian 2-manifold $(M,g)$ of negative Euler characteristic and investigate its properties. We show that $\phi$ always admits a dominated splitting and identify special cases in which $\phi$ is Anosov. In particular, starting from holomorphic differentials of fractional degree, we produce novel examples of Anosov flows on suitable roots of the unit tangent bundle of $(M,g)$.

Projective manifolds whose tangent bundle is Ulrich

Abstract: In this article, we give numerical restrictions on the Chern classes of Ulrich bundles on higher-dimensional manifolds, which are inspired by the results of Casnati in the case of surfaces. As a by-product, we prove that the only projective manifolds whose tangent bundle is Ulrich are the twisted cubic and the Veronese surface. Moreover, we prove that the cotangent bundle is never Ulrich.

Simplicity of tangent bundles of smooth horospherical varieties of Picard number one

Abstract: Recently, Kanemitsu has discovered a counterexample to the long-standing conjecture that the tangent bundle of a Fano manifold of Picard number one is (semi)stable. His counterexample is a smooth horospherical variety. There is a weaker conjecture that the tangent bundle of a Fano manifold of Picard number one is simple. We prove that this weaker conjecture is valid for smooth horospherical varieties of Picard number one. Our proof follows from the existence of an irreducible family of unbendable rational curves whose tangent vectors span the tangent spaces of the horospherical variety at general points.

Volume bounds for the canonical lift complement of a random geodesic

Abstract: Given a filling primitive geodesic loop in a closed hyperbolic surface one obtains a hyperbolic three-manifold as the complement of the loop's canonical lift to the projective tangent bundle. In this paper we give the first known lower bound for the volume of these manifolds in terms of the length of generic loops. We show that estimating the volume from below can be reduced to a counting problem in the unit tangent bundle and solve it by applying an exponential multiple mixing result for the geodesic flow.