核心的余维数为1的具非负曲率完备非紧黎曼流形

利用G .Perelman证明“核心猜想”的思想证明了对n维完备非紧具非负曲率的黎曼流形 ,若其核心之维数是n - 1,则该流形可等距分裂为S×R .其中S为该流形的核心 .     

Nonnegative curvature and cobordism type

We show that in each dimension n = 4k, k≥ 2, there exist infinite sequences of closed simply connected Riemannian n-manifolds with nonnegative sectional curvature and mutually distinct oriented cobordism type. W. Tuschmann’s research was supported in part by a DFG Heisenberg Fellowship.     

Nonnegatively Curved Metrics on S 2 × ℝ2

We classify the complete metrics of nonnegative sectional curvature on M ² × ², where M ² is any compact 2-manifold.     

ON A KAHLER VERSION OF CHEEGER-GROMOLL-PERELMAN＇S SOUL THEOREM

In this note, we will prove a Kahler version of Cheeger-Gromoll-Perelman＇s soul theorem, only assuming the sectional curvature is nonnegative and bisectional curvature is positive at one point.     

When is a Riemannian submersion homogeneous?

We study the structure of the most common type of Riemannian submersions, namely those whose fibers are given by the orbits of an isometric group action on a Riemannian manifold. Special emphasis is given to the case where the ambient space has nonnegative curvature. The first author was supported by research grant MTM2004-04794-MEC. Most of this work was done during a visit of the second author to Madrid, financed in part by funds of the aforementioned grant.     

DARBOUXEQUATIONS AND ISOMETRIC EMBEDDING OF RIEMANNIAN MANIFOLDS WITHNONNEGATIVE CURVATURE INR

50.IntroductionAsDarbouxpointedout,theisometricembeddingoftwodimensionalRiemannianmanifoldsinR3leadstosolveanonlinearpartialdifferentialequationofMongeAmperetypewhereVZz=(iii--r;zk)denotestheHessianofzwithrespecttothegivensmoothmetricg=gijdu'duidefinedonfi,g'Jtheinverseofthemetrictensorandkthecurvatureofthemetricg.Indeed,fromtheGaussequationsoftherequiredisometricembedding/~(x,igz),wherefitjarethecoefficientsofitssecondfundamentalformandacisitsnormal,computingtheinnerproductsofthelastexpre…     

Hardy and Rellich type inequalities on complete manifolds

We prove some Hardy and Rellich type inequalities on complete noncompact Riemannian manifolds supporting a weight function which is not very far from the distance function in the Euclidean space.     

On a kähler version of cheeger-gromoll-perelman's soul theorem

In this note, we will prove a Kähler version of Cheeger-Gromoll-Perelman's soul theorem, only assuming the sectional curvature is nonnegative and bisectional curvature is positive at one point.     

Stability of O(p + 1) × O(p + 1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space

Equivariant geometry methods are used to study and classify zero scalarcurvature O(p + 1) × O(p + 1)-invariant hypersurfaces in ^{2p + 2}with p > 1. First of all the O(p + 1) × O(p + 1)-invariant hypersurfaces are classified according to their profile curves, and it is shown that there are complete and embedded examples. Next the Morse index of the complete examples is studied, in particular, it is proved that there exist globally stable examples.These stable examples provide counter-examples, in odddimensions greater than or equal to nine, to a Bernstein-type conjecture in the stable class, for immersions with zero scalar curvature.