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 the stability of tangent bundle on double coverings

Let Y be a smooth projective surface defined over an algebraically closed field k with char k≠2, and let π : X →Y be a double covering branched along a smooth divisor. We show that y_X is stable with respect to π~*H if the tangent bundle y_Y is semi-stable with respect to some ample line bundle H on Y.     

Stability of Frobenius pull-backs of tangent bundles and generic injectivity of Gauss maps in positive characteristic

For a smooth projective variety X of dimension n in a projective space defined over an algebraically closed field k, the Gauss mapis a morphism from X to the Grassmannian of n-plans in sending to the embedded tangent space .The purpose of this paper is to prove the generic injectivity of Gauss mapsin positive characteristic for two cases; (1) weighted complete intersectionsof dimension of general type; (2) surfaces or 3-folds with -semistable tangent bundles; based on a criterion of Kaji by looking atthe stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman--Piene is true in case X is of general type of dimension . The second result is a generalization of the injectivity for curves.     

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 .

     

一般Hopf曲面上向量丛的结构

得到了非主Hopf曲面上连续复向量丛全纯结构的存在性及可滤性问题的充要条件.     

Conformal vector fields on tangent bundle of a Riemannian manifold

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM,G).     

Existence of homogeneous vectors on the fiber space of the tangent bundle

Let M = G/K be a homogeneous differentiable manifold. We consider the homogeneous bundle = (G, π, G/K, K) and the tangent bundle τ _G/K of M = G/K, and give some results about the existence of homogeneous vectors on the fiber space of τ _G/K, for both cases of G semisimple and weakly semisimple.      

Flow prolongation of some tangent valued forms

We study the prolongation of semibasic projectable tangent valued k-forms on fibered manifolds with respect to a bundle functor F on local isomorphisms that is based on the flow prolongation of vector fields and uses an auxiliary linear r-th order connection on the base manifold, where r is the base order of F. We find a general condition under which the Frölicher-Nijenhuis bracket is preserved. Special attention is paid to the curvature of connections. The first order jet functor and the tangent functor are discussed in detail. Next we clarify how this prolongation procedure can be extended to arbitrary projectable tangent valued k-forms in the case F is a fiber product preserving bundle functor on the category of fibered manifolds with m-dimensional bases and local diffeomorphisms as base maps.     

On decomposition of strongly quasi-frobenius rings

We prove that the Nakayama permutation of a strongly quasi-Frobenius ring gives rise to a ring decomposition of the ring.     

Embeddings of the base and bundle isomorphisms

Let π_i :E_i→M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections s_i :M→E_i. Suppose that U is an open neighborhood of s₁(M) in E₁ and F :U→E₂ a smooth embedding so that π₂Fs₁ :M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E₁ and the induced bundle f^*E₂ are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<kn we give examples of nonisomorphic oriented bundles E₁ and E₂ of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E₁ and f^*E₂ are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.     

四元数矢丛的辛Pontryagin示性式和它的积分公式

设ρ（Ｍ）是微分流形Ｍ上的左四元数矢丛，文中给出ρ（Ｍ）的辛Ｐｏｎｔｒｙａｇｉｎ示性类和辛Ｐｏｎｔｒｙａｇｉｎ示性式的定义，并给出说明它们之间的联系的积分公式。     

The jumping phenomenon of the dimensions of cohomology groups of tangent sheaf

Let X be a compact complex manifold. Consider a small deformation π: X → B of X, the dimensions of the cohomology groups of tangent sheaf H^q(X_t, TX_t) may vary under this deformation. This article studies such phenomena by studying the obstructions to deform a class in H^q(X, TX) with parameter t and gets a formula for the obstructions.     

Geometry of the Second-Order Tangent Bundles of Riemannian Manifolds

Let (M,g) be an n-dimensional Riemannian manifold and T2M be its secondorder tangent bundle equipped with a lift metric (g).In this paper,first,the authors construct some Riemannian almost product structures on (T2M,(g)) and present some results concerning these structures.Then,they investigate the curvature properties of (T2M,(g)).Finally,they study the properties of two metric connections with nonvanishing torsion on (T2 M,(g)):The H-lift of the Levi-Civita connection of g to T2 M,and the product conjugate connection defined by the Levi-Civita connection of (g) and an almost product structure.     

On almost complex structures on tangent bundles

Summary Let <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>(M,J,g)$ be a K\&quot;ahler--Norden manifold. Using the notions of the horizontal and vertical lifts, a class of almost complex structures $\widetilde J$ is defined on the tangent bundle $T\!M$, and necessary and sufficient conditions for such a structure to be integrable (complex) are described. Next, a class of pseudo-Riemannian metrics $\widetilde g$ of Norden type is defined on $T\!M$, for which $\widetilde J$ is an antiisometry. Thus, the pair $(\widetilde J,\widetilde g)$ becomes an almost complex structure with Norden metric on $T\!M$. It is checked whether the structure $(\widetilde J,\widetilde g)$ is K\&quot;ahler--Norden itself.     

一些流形的复化切丛的平凡性

本文研究一个流形M的复化切丛TcM的平凡性问题。我们证明当M是n维球面S^n或者满足某些条件的2,3或4维流形时,TcM是平凡丛．     

Spinors as automorphisms of the tangent bundle

We show that, on a -manifold endowed with a -structure induced by an almost-complex structure, a self-dual (positive) spinor field is the same as a bundle morphism acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of on tangent vectors, and that the squaring map acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic structures.