共查询到20条相似文献,搜索用时 31 毫秒
1.
We present a classification of complete locally irreducible Riemannian manifolds with nonnegative curvature operator, which admit a nonzero and nondecomposable harmonic form with its square-integrable norm. We prove a vanishing theorem for harmonic forms on complete generic Riemannian manifolds with nonnegative curvature operator. We obtain similar results for closed and co-closed conformal Killing forms. 相似文献
2.
Martin HERRMANN 《Frontiers of Mathematics in China》2016,11(5):1259-1274
We give a survey of results on the construction of and obstructions to metrics of almost nonnegative curvature operator on closed manifolds and results on the cohomology rings of closed, simply-connected manifolds with a lower curvature and upper diameter bound. The latter is motivated by a question of Grove whether these condition imply finiteness of rational homotopy types. This question has answers by F. Fang–X. Rong, B. Totaro and recently A. Dessai and the present author. 相似文献
3.
We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with
infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively
curved metrics.
Received February 11, 2000 / Published online February 5, 2001 相似文献
4.
Thomas H. Wolter 《Geometriae Dedicata》1991,37(3):361-370
Homgeneous manifolds of nonpositive sectional curvature can be identified with a certain class of solvable Lie groups. We determine, which of these groups also admit metrics with nonpositive curvature operator; this class is smaller, but still contains many examples. 相似文献
5.
Qi S. Zhang 《Mathematische Annalen》2000,316(4):703-731
By establishing an optimal comparison result on the heat kernel of the conformal Laplacian on open manifolds with nonnegative
Ricci curvature, (a) we show that many manifolds with positive scalar curvature do not possess conformal metrics with scalar
curvature bounded below by a positive constant; (b) we identify a class of functions with the following property: If the manifold
has a scalar curvature in this class, then there exists a complete conformal metric whose scalar curvature is any given function
in this class. This class is optimal in some sense; (c) we have identified all manifolds with nonnegative Ricci curvature,
which are “uniformly” conformal to manifolds with zero scalar curvature. Even in the Euclidean case, we obtain a necessary
and sufficient condition under which the main existence results in [Ni1] and [KN] on prescribing nonnegative scalar curvature
will hold. This condition had been sought in several papers in the last two decades.
Received: 11 November 1998 / Revised: 7 April 1999 相似文献
6.
Gerard Walschap 《Journal of Geometric Analysis》1992,2(4):373-381
We prove a rigidity theorem for Riemannian fibrations of flat spaces over compact bases and give a metric classification of compact four-dimensional manifolds of nonnegative curvature that admit totally geodesic Riemannian foliations. 相似文献
7.
In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick [27, Theorem 1.17]. 相似文献
8.
We investigate (0, 2)-tensors, which fulfil Codazzi-equations, on closed Riemannian manifolds with nonnegative sectional curvature, and give various applications in global differential geometry. 相似文献
9.
Xiaodong Cao 《Mathematische Annalen》2007,337(2):435-441
In this paper, we show that the eigenvalues of
are nondecreasing under the Ricci flow for manifolds with nonnegative curvature operator. Then we show that the only steady Ricci breather with nonnegative curvature operator is the trivial one which is Ricci-flat. 相似文献
10.
Mingliang Cai 《Annals of Global Analysis and Geometry》1993,11(4):373-385
Cheeger and Gromoll proved that a closed Riemannian manifold of nonnegative Ricci curvature is, up to a finite cover, diffeomorphic to a direct product of a simply connected manifold and a torus. In this paper, we extend this theorem to manifolds of almost nonnegative Ricci curvature. 相似文献
11.
J. Ross 《Inventiones Mathematicae》2006,165(1):153-162
We give examples of smooth manifolds with negative first Chern class which are slope unstable with respect to certain polarisations, and so have Kähler classes that do not admit any constant scalar curvature Kähler metrics. These manifolds also have unstable Hilbert and Chow points. We compare this to the work of Song-Weinkove on the J-flow. 相似文献
12.
The manifolds of flags in the projective planes are among the very few compact manifolds that are known to admit metrics with positive sectional curvature. They also arise as isoparametric hypersurfaces in spheres. We show how their appearance in these two fields is related and study the global geodesic geometry of these spaces in detail.Mathematics Subject Classification (2000):53C20, 53C30 相似文献
13.
Yi SHI . Guanghan LI . Chuarixi WU 《数学年刊B辑(英文版)》2014,35(1):93-100
In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov's theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant. 相似文献
14.
We show that a complete, codimension three submanifold M of nonnegative sectional curvature that isometrically splits as ${\bar{M} \times \mathbb{R}}$ has nonnegative curvature operator. We apply this result to obtain a classification of codimension three nonflat manifolds of nonnegative sectional curvature and infinite fundamental group. 相似文献
15.
Curvature,diameter and betti numbers 总被引:12,自引:0,他引:12
Michael Gromov 《Commentarii Mathematici Helvetici》1981,56(1):179-195
We give an upper bound for the Betti numbers of a compact Riemannian manifold in terms of its diameter and the lower bound
of the sectional curvatures. This estimate in particular shows that most manifolds admit no metrics of non-negative sectional
curvature. 相似文献
16.
Wilderich Tuschmann 《Mathematische Annalen》2002,322(2):413-420
The main results of this note consist in the following two geometric finiteness theorems for diffeomorphism types and homotopy
groups of closed simply connected manifolds:
1. For any given numbers C and D the class of closed smooth simply connected manifolds of dimension which admit Riemannian metrics with sectional curvature bounded in absolute value by $\vert K \vert\le C$ and diameter bounded
from above by D contains at most finitely many diffeomorphism types. In each dimension there exist counterexamples to the preceding statement.
2. For any given numbers C and D and any dimension m there exist for each natural number up to isomorphism always at most finitely many groups which can occur as the k-th homotopy group of a closed smooth simply connected m-manifold which admits a metric with sectional curvature and diameter .
Received: 21 August 1999 / Accepted: 20 April 2001 / Published online: 19 October 2001 相似文献
17.
Let N be a closed connected spin manifold admitting one metric ofpositive scalar curvature. In this paper we use the higher eta-invariant associated to the Dirac operator on N in order to distinguish metrics of positive scalar curvature on N. In particular, we give sufficient conditions, involving 1(N) and dim N, for N to admit an infinite number of metrics of positive scalar curvature that are nonbordant.
Mathematics Subject Classifications (2000) 55N22, 19L41. 相似文献
18.
Jian Zhong PAN Shao Bing WU 《数学学报(英文版)》2006,22(1):23-26
in this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact non-negative curved manifolds admit (complete) metrics with non-negative curvature. 相似文献
19.
Maria Helena Noronha 《Geometriae Dedicata》1993,47(3):255-268
In this paper we study some compact locally conformally flat manifolds with a compatible metric whose scalar curvature is nonnegative, and in particular with nonnegative Ricci curvature. In the last section we study such manifolds of dimension 4 and scalar curvature identically zero. 相似文献
20.
Xu Qian FAN 《数学学报(英文版)》2007,23(4):623-638
We study the Poisson equation on some complete noncompact manifolds with asymptotically nonnegative curvature. We will also study the limiting behavior of the nonhomogeneous heat equation on some complete noncompact manifolds with nonnegative curvature. 相似文献