Examples of non-conjugated holomorphic vector fields and foliations

We consider germs of holomorphic vector fields near the origin of with a saddle-node singularity, and the induced singular foliations. In a previous article we described the invariants addressing the analytical classification of these vector fields. They split into three parts: a formal, an orbital and a tangential component. For a fixed formal class, the orbital invariant (associated to the foliation) was obtained by Martinet and Ramis; we give it an integral representation. We then derive examples of non-orbitally conjugated foliations by the use of a “first-step” normal form, whose first-significative jet is an invariant. The tangential invariant also admits an integral representation, hence we derive explicit examples of vector fields, inducing the same foliation, that are not mutually conjugated. In addition, we provide a family of normal forms for vector fields orbitally equivalent to the model of Poincaré-Dulac.     

反正切Finsler度量与Randers度量射影等价(英文)

本文研究了反正切Finsler度量F=α+εβ+βarctan(β/α)与Randers度量F=α+β射影等价,这里α和α表示流形上的两个黎曼度量,β和β表示流形上的两个非零的1-形式.利用射影等价具有相同的Douglas曲率的性质,获得了这两类度量射影等价的充要条件.     

Self-intersecting geodesics and entropy of the geodesic flow

Given a symmetric Finsler metric on T^2 whose geodesic flow has zero topological entropy, we show that the lift in the universal covering R^2 →T^2 of any closed geodesic on T^2 must be an embedded curve in R^2.     

Transversal infinitesimal automorphisms of harmonic foliations on complete manifolds

Summary In this paper we study infinitesimal automorphisms of finiteL ²-norm for harmonic Riemannian and Kähler foliations admitting a complete bundle-like metric. The results generalize facts established recently in the compact case.1980 Mathematics Subject Classification: Primary 57 R 30, Secondary 58 E 20.Work supported in part by a grant from the National Science Foundation.     

Bochner-Kodaira Techniques on K¨ahler Finsler Manifolds

A horizontal Hodge Laplacian operator $\square_{\mathcal {H}}$ is defined for Hermitian holomorphic vector bundles over PTM on K¨ahler Finsler manifold, and the expression of $\square_{\mathcal {H}}$ is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection. The vanishing theorem is obtained by using the $\partial_{\mathcal {H}}\ov{\partial}_{\mathcal {H}}$-method on K¨ahler Finsler manifolds.     

A class of Finsler metrics of scalar flag curvature

It is known that every locally projectively flat Finsler metric is of scalar flag curvature. Conversely, it may not be true. In this paper, for a certain class of Finsler metrics, we prove that it is locally projectively flat if and only if it is of scalar flag curvature. Moreover, we establish a class of new non-trivial examples.     

Projective vector fields on Finsler manifolds

In this paper, we give the equation that characterizes projective vector fields on a Finsler manifold by the local coordinate. Moreover, we obtain a feature of the projective fields on the compact Finsler manifold with non-positive flag curvature and the non-existence of projective vector fields on the compact Finsler manifold with negative flag curvature. Furthermore, we deduce some expectable, but non-trivial relationships between geometric vector fields such as projective, affine, conformal, homothetic and Killing vector fields on a Finsler manifold.     

Characterization of Certain Holomorphic Geodesic Cycles on Quotients of Bounded Symmetric Domains in terms of Tangent Subspaces

Let be an irreducible bounded symmetric domain and Aut() be a torsion-free discrete group of automorphisms, X /. We study the problem of algebro-geometric and differential-geometric characterizations of certain compact holomorphic geodesic cycles S X. We treat special cases of the problem, pertaining to a situation in which S is a compact holomorphic curve, and to the case where is a classical domain dual to the hyperquadric. In both cases we consider algebro-geometric characterizations in terms of tangent subspaces. As a consequence we derive effective pinching theorems where certain complex submanifolds S X are proven to be totally geodesic whenever their scalar curvatures are pinched between certain computed universal constants, independent of the volume of the submanifold S, giving new examples of the gap phenomenon for the characterization of compact holomorphic geodesic cycles.Research funded by a CERG grant from Research Grants Council of Hong Kong.