共查询到20条相似文献,搜索用时 31 毫秒
1.
S. Haesen 《Monatshefte für Mathematik》2007,152(4):303-314
Surfaces with positive definite second fundamental form in a Riemannian, three-dimensional warped product space are considered.
A formula expressing the Gaussian curvature with respect to this new metric on the surface in terms of the Gaussian and mean
curvature of the first fundamental form is presented. This formula is then used to give some characterizations of compact,
totally umbilical surfaces.
Postdoctoral researcher of the F.W.O. Vlaanderen. 相似文献
2.
Andrzej Derdzinski 《Proceedings of the American Mathematical Society》2006,134(12):3645-3648
Let the Ricci curvature of a compact Riemannian manifold be greater, at every point, than the Lie derivative of the metric with respect to some fixed smooth vector field. It is shown that the fundamental group then has only finitely many conjugacy classes. This applies, in particular, to all compact shrinking Ricci solitons.
3.
Xiaodong Cao 《Journal of Geometric Analysis》2007,17(3):425-433
In this article, we first derive several identities on a compact shrinking Ricci soliton. We then show that a compact gradient
shrinking soliton must be Einstein, if it admits a Riemannian metric with positive curvature operator and satisfies an integral
inequality. Furthermore, such a soliton must be of constant curvature. 相似文献
4.
We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined
Weitzenb?ck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show
how it behaves compared to other known bounds.
Received: 20 April 2001 / Published online: 5 September 2002 相似文献
5.
Matthew Cecil 《Bulletin des Sciences Mathématiques》2009,133(4):383-405
Let W(G) and L(G) denote the path and loop groups respectively of a connected real unimodular Lie group G endowed with a left-invariant Riemannian metric. We study the Ricci curvature of certain finite dimensional approximations to these groups based on partitions of the interval [0,1]. We find that the Ricci curvatures of the finite dimensional approximations are bounded below independent of partition iff G is of compact type with an Ad-invariant metric. 相似文献
6.
Seungsu Hwang 《manuscripta mathematica》2000,103(2):135-142
It is well known that critical points of the total scalar curvature functional ? on the space of all smooth Riemannian structures
of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of ? is restricted to the space of constant scalar curvature metrics, there
has been a conjecture that a critical point is also Einstein or isometric to a standard sphere. In this paper we prove that
n-dimensional critical points have vanishing n− 1 homology under a lower Ricci curvature bound for dimension less than 8.
Received: 12 July 1999 相似文献
7.
Gen-ichi Oshikiri 《manuscripta mathematica》2001,104(4):527-531
In this paper, we extend a result by H. Takagi on the non-existence of mutually commuting and linearly independent Killing
vector fields on positively curved Riemannian manifolds. Further, a kind of “Compact Leaf Theorem” is proved for metric foliations
of closed manifolds with positive sectional curvature.
Received: 26 May 2000 / Revised version: 28 February 2001 相似文献
8.
In this paper, we will introduce the notion of harmonic stability for complete minimal hypersurfaces in a complete Riemannian
manifold. The first result we prove, is that a complete harmonic stable minimal surface in a Riemannian manifold with non-negative
Ricci curvature is conformally equivalent to either a plane R
2 or a cylinder R × S
1, which generalizes a theorem due to Fischer-Colbrie and Schoen [12].
The second one is that an n ≥ 2-dimensional, complete harmonic stable minimal, hypersurface M in a complete Riemannian manifold with non-negative sectional curvature has only one end if M is non-parabolic. The third one, which we prove, is that there exist no non-trivial L
2-harmonic one forms on a complete harmonic stable minimal hypersurface in a complete Riemannian manifold with non-negative
sectional curvature. Since the harmonic stability is weaker than stability, we obtain a generalization of a theorem due to
Miyaoka [20] and Palmer [21].
Research partially Supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science
and Technology, Japan.
The author’s research was supported by grant Proj. No. KRF-2007-313-C00058 from Korea Research Foundation, Korea.
Authors’ addresses: Qing-Ming Cheng, Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga
840-8502, Japan; Young Jin Suh, Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea 相似文献
9.
We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2). 相似文献
10.
M. Simon 《manuscripta mathematica》2000,101(1):89-114
The purpose of this paper is to construct a set of Riemannian metrics on a manifold X with the property that will develop a pinching singularity in finite time when evolved by Ricci flow. More specifically, let , where N
n
is an arbitrary closed manifold of dimension n≥ 2 which admits an Einstein metric of positive curvature. We construct a (non-empty) set of warped product metrics on the non-compact manifold X such that if , then a smooth solution , t∈[0,T) to the Ricci flow equation exists for some maximal constant T, 0<T<∞, with initial value , and
where K is some compact set .
Received: 8 March 1999 相似文献
11.
Lisa DeMeyer 《manuscripta mathematica》2001,105(3):283-310
We study the density of closed geodesics property on 2-step nilmanifolds Γ\N, where N is a simply connected 2-step nilpotent Lie group with a left invariant Riemannian metric and Lie algebra ?, and Γ is a lattice
in N. We show the density of closedgeodesics property holds for quotients of singular, simply connected, 2-step nilpotent Lie
groups N which are constructed using irreducible representations of the compact Lie group SU(2).
Received: 8 November 2000 / Revised version: 9 April 2001 相似文献
12.
Large volume growth and the topology of open manifolds 总被引:2,自引:0,他引:2
Changyu Xia 《Mathematische Zeitschrift》2002,239(3):515-526
In this paper, we study complete noncompact Riemannian manifolds with nonnegative Ricci curvature and large volume growth.
We find some reasonable conditions to insure that this kind of manifolds are diffeomorphic to a Euclidean space or have finite
topological type.
Received: January 4, 2000; in final form: October 31, 2000 / Published online: 19 October 2001 相似文献
13.
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
14.
The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this
invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank,
the usual rank, and the 2-number (which is known to be equal to the Euler-Poincare characteristic in these spaces).
Received: 6 June 2000 / Revised version: 6 August 2001 / Published online: 4 April 2002 相似文献
15.
Rafael López Sebastián Montiel 《Calculus of Variations and Partial Differential Equations》1999,8(2):177-190
We give an existence result for constant mean curvature graphs in hyperbolic space . Let be a compact domain of a horosphere in whose boundary is mean convex, that is, its mean curvature (as a submanifold of the horosphere) is positive with respect to the inner orientation. If H is a number such that , then there exists a graph over with constant mean curvature H and boundary . Umbilical examples, when is a sphere, show that our hypothesis on H is the best possible.
Received July 18, 1997 / Accepted April 24, 1998 相似文献
16.
The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this
invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces).
Received: 6 June 2000 / Published online: 1 February 2002 相似文献
17.
We prove that every locally connected quotient G/H of a locally compact, connected, first countable topological group G by
a compact subgroup H admits a G-invariant inner metric with curvature bounded below. Every locally compact homogeneous space
of curvature bounded below is isometric to such a space. These metric spaces generalize the notion of Riemannian homogeneous
space to infinite dimensional groups and quotients which are never (even infinite dimensional) manifolds. We study the geometry
of these spaces, in particular of non-negatively curved homogeneous spaces.
Dedicated to the memory of A. D. Alexandrov 相似文献
18.
We study the Ricci curvature of a Riemannian metric as a differential operator acting on the space of metrics close (in a
weighted functional spaces topology) to the standard metric of a rank-one noncompact symmetric space. We prove that any symmetric
bilinear field close enough to the standard may be realized as the Ricci curvature of a unique close metric if its decay rate
at infinity (its weight) belongs to some precisely known interval. We also study what happens if the decay rate is too small
or too large. 相似文献
19.
Ricci curvature and the topology of open manifolds 总被引:6,自引:0,他引:6
In this paper, we prove that an open Riemannian n-manifold with Ricci curvature and for some is diffeomorphic to a Euclidean n-space if the volume growth of geodesic balls around p is not too far from that of the balls in . We also prove that a complete n-manifold M with is diffeomorphic to if , where is the volume of unit ball in .
Received 5 May, 1997 相似文献
20.