An Interpolation of Hardy Inequality and Moser–Trudinger Inequality on Riemannian Manifolds with Negative Curvature

Let M be a complete, simply connected Riemannian manifold with negative curvature.We obtain an interpolation of Hardy inequality and Moser–Trudinger inequality on M. Furthermore,the constant we obtain is sharp.     

Musical Isomorphisms and Problems of Lifts

Using the complete lift on tangent bundles, the authors construct the complete lift on cotangent bundles of tensor fields with the aid of a musical isomorphism. In this new framework, the authors have a new intrepretation of the complete lift of tensor fields on cotangent bundles.     

Global existence of smooth solutions to 2D Chaplygin gases on curved space

This paper investigates the smooth solution of 2D Chaplygin gas equations on an asymptotically flat Riemannian manifold. Under the assumption that the initial data are close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of smooth solutions to the Cauchy problem for two‐dimensional flow of Chaplygin gases on curved space. Copyright © 2016 John Wiley & Sons, Ltd.     

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

In this paper we present new results on two‐weight Hardy, Hardy–Poincaré and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.     

Reduction,Linearization, and Stability of Relative Equilibria for Mechanical Systems on Riemannian Manifolds

Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory “commute.” As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory. F. Bullo’s research supported in part by grant CMS 0442041 from the USA National Science Foundation. A.D. Lewis’ research supported in part by grant from the Natural Sciences and Engineering Research Council of Canada.     

关于极大算子的几点注记

本文给出了Hardy-Littlewood极大算子的BMO有界姓的一个新证明。用这个证法,我们考虑了其它由卷积产生的极大算子的BMO有界性。最后,我们把Bennett-Devore-Sharpley的定理推广到具有非负Ricci曲率的完备Riemann流形。     

具有平行平均曲率向量场的H稳定子流形

本文讨论局部对称共形平坦Riemann流形N中的紧致H稳定子流形M,若M具于平行平均曲率向量场,则对M的截面曲率或Ricci曲率加上适当的限制条件后,我们证明了M是N中某全脐点子流形N~(N+1)的全脐点超曲面。     

常曲率黎曼流形中的广义旋转极小超曲面

设S~(n+1)(K_0)是具有正常数截面曲率K_0的n+1维黎曼流形,若n维紧致连通广义旋转流形V~n=V~r×p~2S~(n-r)(K)极小浸入在S~(n+1)(K_0)中,则V~n或是S~(n+1)(K_0)的全测地超曲面S~n(K_0)或是V~r是S~(n+1)(K_0)的r_1(相似文献   

关于广义拟einstein流形

近年来,不少研究都涉及到Ricci张量R_(αβ)能写成的黎曼流形,如[1]—[3]等。当一个黎曼流形(M,α)满足(1),且其中ξ为单位向量时,Adati等称该M为ξ-Einstein流形。显然,这类流形的性质不如Einstein流形。因此,专门研究这类流形的文献还不多见。上述各引文与[5]讨论的均为其特别情形。不难知道,白正国教授在[6]中研究的流形也属于此。为和具有类似于形式(1)的其它相应流形的名称(如“拟脐”,“拟常曲率”等)相一致,作者在[7]中称满足(1)的M为拟Einstein的。并简记为QE(ξ)流形,而称ξ为基本元。在那里,我们得到了这类流形的几何特征和Weyl张量满足的代数特征,并指出了某些QE流形的不存在性。本文的目的则是探讨当(1)中ξ是迷向向量场时相应流形的有关性质。