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.

Geodesic flows on manifolds of negative curvature with smooth horospheric foliations

Abstract: We improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.

On manifolds with trivial logarithmic tangent bundle

Abstract: By a classical result of Wang [14] a connected compact complex manifold X has holomorphically trivial tangent bundle if and only if there is a connected complex Lie group G and a discrete subgroup Γ such that X is biholomorphic to the quotient manifold G/Γ. In particular X is homogeneous. If X is Kähler, G must be commutative and the quotient manifold G/Γ is a compact complex torus. The purpose of this note is to generalize this result to the noncompact Kähler case. Evidently, for arbitrary non-compact complex manifold such a result can not hold. For instance, every domain over C has trivial tangent bundle, but many domains have no automorphisms. So we consider the “open case” in the sense of Iitaka ([7]), i.e. we consider manifolds which can be compactified by adding a divisor. Following a suggestion of the referee, instead of only considering Kähler manifolds we consider manifolds in class C as introduced in [5]. A compact complex manifold X is said to be class in C if there is a surjective holomorphic map from a compact Kähler manifold onto X. Equivalently, X is bimeromorphic to a Kähler manifold ([13]). For example, every Moishezon manifold is in class C. We obtain the following characterization:

Moser–Trudinger inequalities of vector bundle over a compact Riemannian manifold of dimension 2

Abstract: Let (M,g) be a two-dimensional compact Riemannian manifold. In this paper, we use the method of blowing up analysis to prove several Moser–Trudinger type inequalities for vector bundle over (M,g). We also derive an upper bound of such inequalities under the assumption that blowing up occurs.

HARMONIC SECTIONS OF TANGENT BUNDLES EQUIPPED WITH RIEMANNIAN g-NATURAL METRICS

Abstract: Let $(M,g)$ be a Riemannian manifold. When $M$ is compact and the tangent bundle $TM$ is equipped with the Sasaki metric $g^s$, the only vector fields which define harmonic maps from $(M,g)$ to $(TM,g^s)$, are the parallel ones. The Sasaki metric, and other well known Riemannian metrics on $TM$, are particular examples of $g$-natural metrics. We equip $TM$ with an arbitrary Riemannian $g$-natural metric $G$, and investigate the harmonicity of a vector field $V$ of $M$, thought as a map from $(M,g)$ to $(TM,G)$. We then apply this study to the Reeb vector field and, in particular, to Hopf vector fields on odd-dimensional spheres.