Geodesic Mappings of Equiaffine and Ricci Symmetric Spaces

Mathematical Notes -     

Riesz Means on Compact Riemannian Symmetric Spaces

We study approximation properties of the Riesz means on compact symmetric spaces of rank one. To do so we establish equivalences between the Riesz means and Peetre K-moduli and estimate the weak type and the uniform approximation of the Riesz means at the critical index. The relations between the Riesz means and the best approximation as well as the Cesàro means are also considered.     

局部共形对称黎曼流形的孤立现象

讨论了局部共形对称的封闭黎曼流形,证明了黎曼曲率张量模长的一个拼挤定理.当M是局部共形平坦流形时,得到了曲率张量模长的最佳拼挤常数,并确定了达到该值的黎曼流形.     

Some Remarks on Complex Crowns of Riemannian Symmetric Spaces

We discuss several results and problems in which complex geometric constructions appear in the geometry of real pseudo Riemannian symmetric spaces in connection with analysis on such spaces.     

Isomorphisms of Sobolev Spaces on Riemannian Manifolds and Quasiconformal Mappings

We prove that a measurable mapping of domains in a complete Riemannian manifold induces an isomorphism of Sobolev spaces with the first generalized derivatives whose summability exponent equals the (Hausdorff) dimension of the manifold if and only if the mapping coincides with some quasiconformal mapping almost everywhere.

     

Four-Dimensional Pseudo-Riemannian Generalized Symmetric Spaces Which are Algebraic Ricci Solitons

We classify, up to isometry, non-symmetric simply-connected four-dimensional pseudo-Riemannian generalized symmetric spaces which are algebraic Ricci solitons. It turns out that those of Cerny–Kowalski’s types A, C and D are algebraic Ricci solitons, whereas those of type B are not. Thus, we give new examples of algebraic Ricci solitons.     

A Reconstruction Theorem for Riemannian Symmetric Spaces of Noncompact Type

Let f be a rapidly decreasing radial function on a Riemannian symmetric space of noncompact type whose spherical Fourier transform has compact support. We prove a reconstruction theorem which recovers f from the values of an integral operator applied to f on a discrete subset. When G/K is of the complex type we prove a sampling formula recovering f from its own values on a discrete subset. We give explicit results for three dimensional hyperbolic space.     

The Partial Positivity of the Curvature in Riemannian Symmetric Spaces

In this paper, the partial positivity （resp., negativity） of the curvature of all irreducible Riemannian symmetric spaces is determined. From the classifications of abstract root systems and maximal subsystems, the author gives the calculations for symmetric spaces both in classical types and in exceptional types.     

Some Equivalent Norms in Sobolev-Besov Spaces on Symmetric Riemannian Manifolds

In this paper we introduce local means on Riemannian symmetricmanifolds of the noncompact type corresponding to the Laplace-Beltramioperator, and investigate equivalent norms in the Sobolev andBesov spaces defined via these means.     

非紧致一秩Riemann对称空间上的中心极限定理

我们在本文中研究非紧致一秩Ｒｉｅｍａｎｎ对称空间上初等球函数的渐近表示,并利用Ｌｏｈｏｕｅ Ｎ．和Ｒｙｃｈｎｅｒ Ｔｈ．得到的热核表达式,建立起这类空间上的非欧中心极限定理,所得结果包含了Ｔｅｒｒａｓ的定理作为其特例．     

非紧致一秩Rienann对称空间上的中心极限定理

我们在本文中研究非紧致一秩Riemann对称空间上初等球函数的渐近表示,并利用LohoueN.和RychnerTh.得到的热核表达式,建立起这类空间上的非欧中心极限定理,所得结果包含了Terras的定理作为其特例.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

Spaces of Riemannian metrics

In this paper, we consider spaces M of Riemannian metrics on a closed manifold M. In the case where the manifold M is equipped with a symplectic or contact structure, we consider spaces AM of associated metrics. We study geometric and topological properties of these spaces and Riemannian functionals on spaces of metrics. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 31, Geometry, 2005.     

Smoothing Riemannian metrics with Ricci curvature bounds

We prove that Riemannian metrics with an absolute Ricci curvature bound and a conjugate radius bound can be smoothed to having a sectional curvature bound. Using this we derive a number of results about structures of manifolds with Ricci curvature bounds. The authors were supported in part by NSF Grant. The first author was also supported in part by Alfred P. Sloan Fellowship This article was processed by the author using the LATEX style filecljourl from Springer-Verlag.     

Ricci Deformation of the Metric on Riemannian Orbifolds

In this note we generalize the Huisken’s (J Diff Geom 21:47–62, 1985) result to Riemannian orbifolds. We show that on any n-dimensional (n ≥ 4) orbifold of positive scalar curvature the metric can be deformed into a metric of constant positive curvature, provided the norm of the Weyl conformal curvature tensor and the norm of the traceless Ricci tensor are not large compared to the scalar curvature at each point, and therefore generalize 3-orbifolds result proved by Hamilton [Three- orbifolds with positive Ricci curvature. In: Cao HD, Chow B, Chu SC, Yau ST (eds) Collected Papers on Ricci Flow, Internat. Press, Somerville, 2003] to n-orbifolds (n ≥ 4).     

Conformal Ricci flow on asymptotically hyperbolic manifolds

In this article, we study the short-time existence of conformal Ricci flow on asymptotically hyperbolic manifolds. We also prove a local Shi's type curvature derivative estimate for conformal Ricci flow.     

Isometric immersions of Riemannian Spaces in Euclidean Spaces

Questions of the theory of isometric immersions of Riemannian spaces in Euclidean spaces beginning with the very first results onthis topic and also results on immersions of pseudo-Riemannian spaces in pseudo-Euclidean spaces and applications of the theory of immersions in the general theory of relativity are considered.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 15, pp. 173–211, 1977.