Let （M, g） be an n-dimensional Riemannian manifold and T＊M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki ...Let （M, g） be an n-dimensional Riemannian manifold and T＊M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compati-ble paracomplex structures on T＊M. Second, we construct locally decomposable Golden Riemannian structures on T＊M . Finally we investigate curvature properties of T＊M .展开更多
Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle ...Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle on X, and ρ：GLn（k） → GLm（k） a rational GLn（k）-representation of degree at most d such that ρ maps the radical R（GLn（k）） of GLn（k） into the radical R（GLm（k）） of GLm（k）. We show that if FXN＊（E） is semistable for some integer N ≥ max0 〈 r 〈 m （rm） · logp（dr）, then the induced rational GLm（k）-bundle E（GLm（k）） is semistable. As an application, if dim X=n, we get a sufficient condition for the semistability of Frobenius direct image FX＊（ρ＊（ΩX1））, where ρ＊（ΩX1） is the vector bundle obtained from ΩX1 via the rational representation ρ.展开更多
In this paper.we discuss Lagrangian vector field on Kahler manifold and use it to describe and solve some problem in Newtonican and Lagrangian Mechanics on Kahler Manifold.
In [2-4], symplectic schemes of arbitrary order are constructed by generating functions. However the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what ...In [2-4], symplectic schemes of arbitrary order are constructed by generating functions. However the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what circumstance the construction of generating functions will be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations. This paper will begin with the study of Darboux transformation, then in section 2, a normalization Darboux transformation will be defined naturally. Every symplectic scheme which is constructed from Darboux transformation and compatible with the Hamiltonian equation will satisfy this normalization condition. In section 3, we will study transformation properties of generator maps and generating functions. Section 4 will be devoted to the study of the relationship between the invariance of generating functions and the generator maps. In section 5, formal symplectic erengy of symplectic schemes are presented.展开更多
文摘Let （M, g） be an n-dimensional Riemannian manifold and T＊M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compati-ble paracomplex structures on T＊M. Second, we construct locally decomposable Golden Riemannian structures on T＊M . Finally we investigate curvature properties of T＊M .
基金Supported by National Natural Science Foundation of China(Grant No.11501418)Shanghai Sailing Program(Grant No.15YF1412500)
文摘Let k be an algebraically closed field of characteristic p 〉 0, X a smooth projective variety over k with a fixed ample divisor H, FX：X → X the absolute Frobenius morphism on X. Let E be a rational GLn（k）-bundle on X, and ρ：GLn（k） → GLm（k） a rational GLn（k）-representation of degree at most d such that ρ maps the radical R（GLn（k）） of GLn（k） into the radical R（GLm（k）） of GLm（k）. We show that if FXN＊（E） is semistable for some integer N ≥ max0 〈 r 〈 m （rm） · logp（dr）, then the induced rational GLm（k）-bundle E（GLm（k）） is semistable. As an application, if dim X=n, we get a sufficient condition for the semistability of Frobenius direct image FX＊（ρ＊（ΩX1））, where ρ＊（ΩX1） is the vector bundle obtained from ΩX1 via the rational representation ρ.
文摘In this paper.we discuss Lagrangian vector field on Kahler manifold and use it to describe and solve some problem in Newtonican and Lagrangian Mechanics on Kahler Manifold.
文摘In [2-4], symplectic schemes of arbitrary order are constructed by generating functions. However the construction of generating functions is dependent on the chosen coordinates. One would like to know that under what circumstance the construction of generating functions will be independent of the coordinates. The generating functions are deeply associated with the conservation laws, so it is important to study their properties and computations. This paper will begin with the study of Darboux transformation, then in section 2, a normalization Darboux transformation will be defined naturally. Every symplectic scheme which is constructed from Darboux transformation and compatible with the Hamiltonian equation will satisfy this normalization condition. In section 3, we will study transformation properties of generator maps and generating functions. Section 4 will be devoted to the study of the relationship between the invariance of generating functions and the generator maps. In section 5, formal symplectic erengy of symplectic schemes are presented.
