Unit tangent bundle

About: Unit tangent bundle is a research topic. Over the lifetime, 1056 publications have been published within this topic receiving 15845 citations.

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).

Legendre transformation and lifting of multi-vectors

Abstract: This paper has three objectives. First to recall the link between the classical Legendre-Fenschel transformation and a useful isomorphism between 1-jets of functions on a vector bundle and on its dual. As a particular consequence we obtain the classical isomorphism between the cotangent bundle of the tangent bundle $T^*TM$ and the tangent bundle of the cotangent bundle $TT^*M$ of any manifold $M.$ Secondly we show how to use this last isomorphism to construct the lifting of any contravariant tensor field on a manifold $M$ to the tangent bundle $TM$ which generalizes the classical lifting of vector fields. We also show that, in the antisymmetric case, this lifting respects the Schouten bracket. This gives a new proof of a recent result of Crainic and Moerdijk. Finally we give an application to the study of the stability of singular points of Poisson manifold and Lie algebroids.

When is an Anosov flow geodesic

Abstract: Let X, H+, H− be vector fields tangent, respectively, to an Anosov flow and its expanding and contracting foliations in a compact three-dimensional manifold, with γ, α+, α− one forms dual to them. If α+([H+, H−]) = α−([H+, H−]) and γ([H+, H−]) = α−([X, H−]) − α+([X, H+]), then the manifold has the structure of the unit tangent bundle of a Riemannian orbifold with geodesic flow field X.

Uniruled varieties with split tangent bundle

Abstract: Beauville asked if a compact Kahler manifold with split tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic foliations to study this problem for projective uniruled varieties. In particular we obtain an affirmative answer for rationally connected varieties in any dimension and uniruled varieties in dimension 4.

Geometric structure for the tangent bundles of direct limit manifolds

Abstract: We equip the direct limit of tangent bundles of paracompact finite dimensional manifolds with a structure of convenient vector bundle with structural group GL(∞,R) = lim→ GL(R n ).