局部对称的黎曼流形中的极小子流形

主要研究了局部对称的黎曼流形中的定向紧致无边极小子流形的内蕴刚性问题,利用一个矩阵不等式,得到了这类子流形的一个刚性定理．所得结果部分改进了已有的一个结论．     

局部对称共形平坦黎曼流形中紧致子流形的一个刚性定理

本文研究局部对称共形平坦黎曼流形N^n p(p≥2）中具有平行平均曲率向量的紧致子流形M^n的余维可约性问题,在n≥8的条件下得到了最佳拼挤常数。     

Minimal submanifolds in a Riemannian manifold with pinched positive curvature

In this paper, we prove the following result: LetM be ann-dimensional compact curvature-invariant minimal subinanifoid immersed in a(> 2n–2/5n–2)- pinched Riemannian manifoldN. Denote by the second fundamental form ofM. If ¦¦²<3/9((5n–2)–2(n–1)), thenM is totally geodesic. This result generalizes the Simons pinching theorem.Supported by the JSPS postdoctoral fellowship and NSF of China.     

Fields of symmetric tensors on a compact Riemannian manifold

Translated from Matematicheskii Zametki, Vol. 52, No. 4, pp. 85–88, October, 1992.     

Rigidity theorem for harmonic maps from Riemannian manifold to Grassmannian manifold

We first study the Grassmannian manifoldG _n (R^n+p)as a submanifold in Euclidean space ⁿ (R ^n+p). Then we give a local expression for each map from Riemannian manifoldM toG ⁿ (R ^n+p) ⁿ (R ^n+p), and use the local expression to establish a formula which is satisfied by any harmonic map fromM toG _n (R ^n+p). As a consequence of this formula we get a rigidity theorem.     

The metric geometry of the manifold of Riemannian metrics over a closed manifold

We prove that the L ² Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L ² metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.     

Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold

We show that for every closed Riemannian manifold X there exists a positive number¶ $ \varepsilon_0 > 0 $ \varepsilon_0 > 0 such that for all 0< e\leqq e₀ \varepsilon \leqq \varepsilon_0 there exists some¶ $ \delta > 0 $ \delta > 0 such that for every metric space Y with Gromov-Hausdorff distance to X less than¶ d \delta the geometric e \varepsilon -complex |Y_e| |Y_\varepsilon| is homotopy equivalent to X.¶ In particular, this gives a positive answer to a question of Hausmann [4].     

Complete space-like submanifolds in locally symmetric semi-definite spaces

The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized     

局部对称黎曼流形中的超曲面

研究了局部对称黎曼流形中具有常平均曲率超曲面的几何性质,得到了关于其第二基本形式模长平方的拼挤定理.     

The length of a shortest geodesic net on a closed Riemannian manifold

In this paper we will estimate the smallest length of a minimal geodesic net on an arbitrary closed Riemannian manifold Mⁿ in terms of the diameter of this manifold and its dimension. Minimal geodesic nets are critical points of the length functional on the space of immersed graphs into a Riemannian manifold. We prove that there exists a minimal geodesic net that consists of m geodesics connecting two points p,q∈Mⁿ of total length ≤md, where m∈{2,…,(n+1)} and d is the diameter of Mⁿ. We also show that there exists a minimal geodesic net with at most n+1 vertices and geodesic segments of total length .These results significantly improve one of the results of [A. Nabutovsky, R. Rotman, The minimal length of a closed geodesic net on a Riemannian manifold with a nontrivial second homology group, Geom. Dedicata 113 (2005) 234-254] as well as most of the results of [A. Nabutovsky, R. Rotman, Volume, diameter and the minimal mass of a stationary 1-cycle, Geom. Funct. Anal. 14 (4) (2004) 748-790].     

Totally geodesic submanifolds in compact Riemannian symmetric spaces and their stability

In this paper, we determine a large class of totally geodesic submanifolds of a compact Riemannian symmetric space. The stability of these submanifolds in their ambient space is also determined.     

Totally geodesic Riemannian foliations with locally symmetric leaves

We prove the arithmeticity of totally geodesic Riemannian foliations, with a dense leaf, on complete finite volume Riemannian manifolds when the leaves are isometrically covered by an irreducible symmetric space of noncompact type and rank at least 2. To cite this article: R. Quiroga-Barranco, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On locally symmetric vector fields on Riemannian manifolds

It is shown that if ann-dimensional (n≧3) Riemannian manifold admitsr≧2 locally symmetric vector fields (LSVF's), then it is aV(k)-space. In particular, ifr=n−1 then the manifold is a space of constant curvature. In the case of a 3-dimensional Riemannian manifold a close connection between LSVF's and eigenvectors of the Ricci tensor is found.     

局部对称流形中具有常平均曲率的完备超曲面

讨论了局部对称黎曼流形中具有常平均曲率的完备超曲面的性质,通过Laplace算子的计算,得到一个关于第二基本形式模长平方S的拼挤定理,推广了已有的结果．     

关于局部对称空间中极小子流形的n个整体拼挤定理

研究局部对称δ-拼挤黎曼流形中紧致的极小子流形,给出了若干个整体的拼挤定理,推广了S.S.Chern,M.do Carm o,S.K obayash i及S.T.Y au相应的结果.     

Polypotentials on a Riemannian manifold

Polyharmonic functions are considered on open sets in a Riemannian manifold R and their potential-theoretic properties are studied using the notion of complete m-potentials. Also one obtains here some characterizations of domains in R on which such complete m-potentials exist.     

Curve shortening in a Riemannian manifold

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and , then for every n > 0, . Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S ¹, if γ₀ is a ramp, then we have a global flow which converges to a closed geodesic in C ^∞ norm. As an application, we prove the theorem of Lyusternik and Fet.      

Spherical-type hypersurfaces in a Riemannian manifold

We extend some rigidity results of Aleksandrov and Ros on compact hypersurfaces inR ⁿ to more general ambient spaces with the aid of the notion of almost conformal vector fields. These latter, at least locally, always exist and allow us to find interesting integral formulas fitting our purposes.     

An exhaustion of locally symmetric spaces by compact submanifolds with corners

LetX be a Riemannian symmetric space of noncompact type and rank2 and let be a non-uniform, irreducible lattice. On the locally symmetric quotientV=/X we construct an exhaustion functionh:V[0,) whose sublevel sets {hs} are compact submanifolds ofV with corners. The top dimensional boundary faces of {hs} are parts of certain horospheres that join together at the corners. It can be shown that actually {hs} is a submanifold with corners isomorphic to the Borel-Serre compactification ofV.Oblatum 2-VIII-1993 & 19-XII-1994