Kato Space for Dirichlet Forms

We study the space of Kato measures relative to a Dirichlet form and we prove that a local solution of a problem relative to a Kato measure is locally continuous. Moreover if the measure of an intrinsic ball is equivalent to a power of the radius we prove also that the density of the form relative to a local solution is locally a Kato measure.     

从不定度量空间形式到不定度量空间形式的等距浸入

设Msm(c)是等距浸入在2m-1维不定度量空间形式Ns2m-1((?))(c<(?))中的m维不定度量空间形式．若Msm(c)是极小的,我们证明Msm必定是有同一个指标s的2m-1维伪球面中的平坦子流形．我们还用孤立子理论给出了Ns2m-1中平坦的指标为s的子流形与系统之间的对应．     

Isospectrality and -manifold groups

The Chern-Simons and -invariants are closely related invariants of a Riemannian -manifold. A difference in their behavior under taking a finite covering space is exploited to give an obstruction to a group being the fundamental group of a closed -dimensional manifold.

     

Ribaucour Transformations for Hypersurfaces in Space Forms

The theory of Ribaucour transformations for hypersurfaces in space forms is established. For any such hypersurface M, that admits orthonormal principal vector fields, it was shown the existence of a totally umbilic hypersurface locally associated to M by a Ribaucour transformation. A method of obtaining linear Weingarten surfaces in a three-dimensional space form is provided. By applying the theory, a new one-parameter family of complete constant mean curvature (cmc) surfaces in the unit sphere, locally associated to the flat torus, is obtained. The family contains a class of complete cmc cylinders in the sphere. In particular, one gets a family of complete minimal surfaces and minimal cylinders, locally associated to the Clifford torus.Mathematics Subject Classifications (2000): 53C20.     

球面型空间中的度量平均

本文首先建立球面型空间中度量平均的概念，其次讨论度量平均过程中一些几何不变量之间的关系，最后举例说明度量平均在解决球面型空间中几何极值问题时的应用．     

不定度量空间型到不定度量空间型的等距浸入和孤立子理论

我们利用孤立子理论得到了构造有相同指标的不定度量空间型到不定度量空间型的等距浸入的方法．     

Isospectrality and galois projective geometries

We construct a series of pairs of domains in the plane and pairs of surfaces with boundary that are isospectral but not isometric. The construction is based on the existence of finite transformation groups that are spectrally equivalent but not isomorphic. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 660–664, May, 1998. This research was partially supported by the Russian Foundation for Basic Research and by INTAS under joint grant No. 96-01-00043.     

余辛空间形式中的子流形不等式

For submanifolds in a cosymplectic space form tangent to the structure vector field ξ, two important inequalities with Ricci curvature, scalar curvature and the squared mean curvature are obtained. These results are also applied to get corresponding consequences for anti-invariant submanifolds.     

A Characterization for Geodsic Spheres in Space Forms

We use the maximum principle for second-order elliptic operators to establish a sufficient condition for a compact hypersurface in a space form to be a geodesic sphere in terms of a pinching for the s-mean curvature.     

初等图形在球面型空间的实现问题

本文提出并解决了初等图形在球面型空间实现的问题: 预给两两之间度量的几何元素e₁,e₂…,e_l-1,e_l+1,…,e_N(e_i为点或超平面),初等图形{e₁,…,e_l-1,e_l+1,…,e_N}在球面型空间实现的充分必要条件是什么?     

Rigidity for Closed Totally Umbilical Hypersurfaces in Space Forms

In Perez (Thesis, 2011), Perez proved some L ² inequalities for closed convex hypersurfaces immersed in the Euclidean space ? ⁿ⁺¹, and more generally for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this paper, we discuss the rigidity of these inequalities when the ambient manifold is ? ⁿ⁺¹, the hyperbolic space ? ⁿ⁺¹, or the closed hemisphere \(\mathbb{S}_{+}^{n+1}\) . We also obtain a generalization of Perez’s theorem to the hypersurfaces without the hypothesis of non-negative Ricci curvature.     

球面型空间有限点集的两个不等式

运用距离几何的理论与方法，给出了涉及球面型空间有限点集的两个不等式．     

Conformal CMC-Surfaces in Lorentzian Space Forms

Let Q3 be the common conformal compactification space of the Lorentzian space forms R13, S13 and H13. We study the conformal geometry of space-like surfaces in Q3. It is shown that any conformal CMC-surface in Q3 must be conformally equivalent to a constant mean curvature surface in R13, S13 or H13. We also show that if x : M→Q3 is a space-like Willmore surface whose conformal metric g has constant curvature K, then either K = - 1 and x is conformally equivalent to a minimal surface in R13, or K = 0 and x is conformally equivalent to the surface H1(1/(2~(1/2)))×H1(1/(2~(1/2))) in H13.     

Killing Vector Fields and Twistor Forms on Generalized Sasakian Space Forms

We study generalized Sasakian space form M(f ₁, f ₂, f ₃) when (i) the Reeb vector field of the almost contact metric structure is Killing, (ii) the Ricci tensor satisfies Einstein-like conditions and (iii) the fundamental 2-form of the almost contact metric structure is a twistor form.     

Sasakian空间形式中子流形的全实截面曲率(英文)

By using the methods introduced by Chen[Chen Bang-yen,A series of Ka¨hlerian invarianrts and their applications to Khlerian geometry,Beitrge Algebra Geom,2001,42（1）：165-178],we establish some inequalities for invariant submanifolds in a Sasakian space form involving totally real sectional curvature and the scalar curvature.Moreover,we consider the case of equalities.     

Lagrangian Bonnet Problems in Complex Space Forms

In this note we consider Lagrangian Bonnet problem for Lagrangian surfaces in complex space forms. We first give a Bonnet type theorem for conformal Lagran