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.     

Obstructions to nonnegative curvature and rational homotopy theory

We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if lies in the class and is a torus of positive dimension, then ``most' vector bundles over admit no complete nonnegatively curved metrics.

     

Vector bundles with infinitely many souls

We construct the first examples of manifolds, the simplest one being , which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.

     

DARBOUX EQUATIONS AND ISOMETRIC EMBEDDING OF RIEMANNIAN MANIFOLDS WITH NONNEGATIVE CURVATURE IN R

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

Nonnegatively curved vector bundles with large normal holonomy groups

When is a biquotient, we show that there exist vector bundles over with metrics of nonnegative curvature whose normal holonomy groups have arbitrarily large dimension.

     

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 .