CONFORMALLY FLAT AFFINE HYPERSURFACES WITH SEMI-PARALLEL CUBIC FORM

In this paper,we study locally strongly convex affine hypersurfaces with the vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of the affine metric.As a main result,we classify these hypersurfaces as not being of a flat affine metric.In particular,2 and 3-dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic forms are completely determined.

The Submanifolds with Parallel Mean Curvature Vector in a Locally Symmetric and Conformally Flat Riemannian Manifold

In the present paper we obtain the following result: Theorem Let M^R be a compact submanifold with parallel mean curvature vector in a locally symmetric and conformally flat Riemannian manifold N^(n+p)(p>1). If then M^n lies in a totally geodesic submanifold N^(n+1).

On Submanifolds in Locally Symmetric and Conformally Flat Riemannian Manifolds

Let N n+p be an (n+p)-dimensional locally symmetric and conformally flat Riemannian manifold and Mn be an n-dimensional compact submanifold minimally immersed in N n+p . Instead of (n+p)-dimensional unit sphere, we generalize Pinching Theorems about submanifolds in unit sphere and get theorems about submanifolds in locally symmetric and conformally flat Riemannian manifold.

The Minimal Submanifolds in a Locally Symmetric and Conformally Flat Riemannian Manifold

In this paper,we extend two important theorem in[1],[2]to the minimal submanifolds in a Locally symmetric and conformally flat Riemannian mainfold N^(+p).When N^(+p)is a space S_(1)^(+p) of constant curvature,our theorems reduce to the theorems of[1],[2].

RIGIDITY OF COMPACT MINIMAL SUBMANIFOLDS IN A LOCALLY SYMMETRIC AND CONFORMALLY FLAT RIEMANN MANIFOLD

The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method is completely different. Meanwhile,we get better conclusion than that of [3].We also research the Pinching problem for sectional curvature on compact minimal submanifolds in a unit sphere, partially improving the results of S.T.Yan[4].

Conformally flat Lorentzian hypersurfaces in R_1~4 with three distinct principal curvatures

A three dimensional Lorentzian hypersurface x : M_1~3→ R_1~4 is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of R_1~4. Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.

Generic conformally flat hypersurfaces and surfaces in 3-sphere

The aim of this paper is to verify that the study of generic conformally flat hypersurfaces in 4-dimensional space forms is reduced to a surface theory in the standard 3-sphere.The conformal structure of generic conformally flat(local-)hypersurfaces is characterized as conformally flat(local-)3-metrics with the Guichard condition.Then,there is a certain class of orthogonal analytic(local-)Riemannian 2-metrics with constant Gauss curvature-1 such that any 2-metric of the class gives rise to a one-parameter family of conformally flat 3-metrics with the Guichard condition.In this paper,we firstly relate 2-metrics of the class to surfaces in the 3-sphere:for a 2-metric of the class,a 5-dimensional set of(non-isometric)analytic surfaces in the 3-sphere is determined such that any surface of the set gives rise to an evolution of surfaces in the 3-sphere issuing from the surface and the evolution is the Gauss map of a generic conformally flat hypersurface in the Euclidean4-space.Secondly,we characterize analytic surfaces in the 3-sphere which give rise to generic conformally flat hypersurfaces.

Conformally flat, minimal, Lagrangian submanifolds in complex space forms

We investigate n-dimensional(n≥4),conformally flat,minimal,Lagrangian submanifolds of the n-dimensional complex space form in terms of the multiplicities of the eigenvalues of the Schouten tensor and classify those that admit at most one eigenvalue of multiplicity one.In the case where the ambient space is Cn,the quasi umbilical case was studied in Blair(2007).However,the classification there is not complete and several examples are missing.Here,we complete(and extend) the classification and we also deal with the case where the ambient complex space form has non-vanishing holomorphic sectional curvature.

Quasi-con form ally Flat Manifolds with Constant Scalar Curvature

Goldberg and Wu studied a conformally flat manifold M with constant scalar curvature. When the Ricci curvature of M is of bounded below or positive,the conditions of M becoming a constant curvature manifold are obtained. In this paper,we consider conharmonically flat manifolds and quasi conformally flat manifolds with constant saclar curvature. The corresponding results are generalized.

共形平坦的黎曼流形

设M是n>2维连通的微分流形。本文利用微分几何中的Bochner技巧证明了下述定理: 定理A.设M是n>2维紧致、共形平坦的黎曼流形,具常标量曲率,则M是常曲率黎曼流形。文献[1]证明了下述定理:设M是n≥3维紧致、共形平坦的黎曼流形,具有常标置曲率r.如果RiCCi张量的长度小于r/2n-1,则M是常曲率的。 [1]文是用“夹击”（Pinch）Ricci张量的方法证明上述结果的。如定理A所示,在很自然的前提下（微分流形M是连通的）关于Ricci张量的长度的限制可以丢掉。

Vanishing Results for the Cotton Tensor on Gradient Quasi-Einstein Solitons

In this paper we study on gradient quasi-Einstein solitons with a fourth-order vanishing condition on the Weyl tensor.More precisely,we show that for n≥4,the Cotton tensor of any ndimensional gradient quasi-Einstein soliton with fourth order f-divergence free Weyl tensor is flat,if the manifold is compact,or noncompact but the potential function satisfies some growth condition.As corollaries,some local characterization results for the quasi-Einstein metrics are derived.

Conformal Killing Vectors in LRS Bianchi Type Ⅴ Spacetimes

In this note,we investigate conformal Killing vectors(CKVs)of locally rotationally symmetric(LRS)Bianchi type V spacetimes.Subject to some integrability conditions,CKVs up to implicit functions of(t,x)are obtained.Solving these integrability conditions in some particular cases,the CKVs are completely determined,obtaining a classification of LRS Bianchi type V spacetimes.The inheriting conformal Killing vectors of LRS Bianchi type V spacetimes are also discussed.

Gap Theorem on Complete Noncompact Riemannian Manifold

A gap theorem on complete noncompact n-dimensional locally conformally flat Riemannian manifold with nonnegative and bounded Ricci curvature is proved.If there holds the following condition：integral（sk（x0,s）ds= o（log r）） from n=0 to r then the manifold is flat.