g-natural symmetries on tangent bundles

The study of symmetries is a well known research topic in differential geometry with relevant physical interpretations. Given a Riemannian manifold $(M,g)$, we consider pseudo-Riemannian g-natural metrics G on its tangent bundle $TM$ and characterize conformal, homothetic and Killing vector fields of $(TM,G)$ obtained from natural lifts of vector fields of M.     

Low-dimensional projective manifolds with nef tangent bundle in positive characteristic

We classify smooth projective varieties with nef tangent bundle in positive characteristic, when the varieties are surfaces or Fano 3-folds. Furthermore, some related problems will be discussed.     

On twistor Gauss maps of surfaces in 4-spheres

In this paper we study surfaces in S⁴ and their twistor Gauss maps. Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given. We find many examples of nonisotropic harmonic maps from a surface to P ³.Supported by the National Natural Science Foundation of China and the Science Foundation of Zhejiang Province.     

Rigidity and mean curvature flow via harmonic Gauss maps

We investigate properties of harmonic Gauss maps and their applications to Lawson-Osserman’s problem, to the rigidity of space-like submanifolds in a pseudo-Euclidean space and to the mean curvature flow.     

3维双曲空间中曲面的双曲Gauss映照和法Gauss映照

本文导出了3维双曲空间中曲面的双曲Gauss映照和法Gauss映照的关系,发现了一般的曲面由双曲Gauss映照和平均曲率函数唯一确定,并证明了双曲Gauss映照所满足的二阶线性椭圆方程,给出了两种形式的关于双曲Gauss映照的三阶非线性偏微分方程(组)的一个解.     

Mass transport generated by a flow of Gauss maps

Let A⊂Rd, d?2, be a compact convex set and let be a probability measure on A equivalent to the restriction of Lebesgue measure. Let be a probability measure on equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν=μ○T⁻¹ and T=φ⋅n, where is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of φ. Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth φ the level sets of φ are governed by the Gauss curvature flow , where K is the Gauss curvature. As a by-product one can reprove the existence of weak solutions to the classical Gauss curvature flow starting from a convex hypersurface.     

Ample and spanned vector bundles of top Chern number two on smooth projective varieties

The purpose of this paper is to classify ample and spanned vector bundles of top Chern number two on smooth projective varieties of arbitrary dimension defined over an algebraically closed field of characteristic zero.

     

Pseudo-hyperbolic Gauss maps of Lorentzian surfaces in anti-de Sitter space

In this paper, we determine the type numbers of the pseudo-hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the 3-dimensional anti-de Sitter space. Also, we investigate the behavior of type numbers of the pseudo-hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the 3-dimensional anti-de Sitter space. Furthermore, we investigate the type number of the pseudo-hyperbolic Gauss map of one of Lorentzian hypersurfaces of B-scroll type in a general dimensional anti-de Sitter space.     

Sub-bundles of the complexified tangent bundle

We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold. The aim is to understand implications of properties of interest in partial differential equations.

     

The power of the tangent bundle of the real projective space, its complexification and extendibility

We establish the formulas on the power of the tangent bundle of the real projective -space and its complexification , and apply the formulas to the problem of extendibility and stable extendiblity of and .

     

复射影空间中子流形的Gauss映照

本文利用复射影空间到欧氏空间的第一标准嵌入，对于复射影空间的子流形建立了一种广义的Ｇａｕｓｓ映照，并给出了这种广义的Ｇａｕ８ｓ映照是调和映照和相对仿射映照的条件。     

欧氏空间中具有共形Gauss映照的曲面

本文考虑欧氏空间中具有共形Ｇａｕｓｓ映照的曲面．从Ｇａｕｓｓ映照的观点给出了 ｖｅｒｏｎｅｓｅ曲面的一个新特征．     

On the classification of contact metric ‐spaces via tangent hyperquadric bundles

We classify locally the contact metric ‐spaces whose Boeckx invariant is as tangent hyperquadric bundles of Lorentzian space forms.     

The value distribution of the hyperbolic Gauss map

In this paper, we investigate the hyperbolic Gauss map of a complete CMC-1 surface in , and prove that it cannot omit more than four points unless the surface is a horosphere.

     

A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds

Fix a smooth projective 3-fold , , with ample, and . Assume the existence of integers with such that is numerically equivalent to . Let be the moduli scheme of -stable rank 2 vector bundles, , on with and . Let be the number ofits irreducible components. Then .

     

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.     

Tensor Fields of Type (0, 2) on the Tangent Bundle of a Riemannian Manifold

To any (0, 2)-tensor field on the tangent bundle of a Riemannian manifold, we associate a global matrix function. Based on this fact, natural tensor fields are defined and characterized, essentially by means of well-known algebraic results. In the symmetric case, this classification coincides with the one given by Kowalski–Sekizawa; in the skew-symmetric one, it does with that obtained by Janyka.     

Stability and integrability of horizontally conformal maps and harmonic morphisms

In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.