Homogeneous contact Riemannian three-manifolds
Homogeneous contact Riemannian three-manifolds
复制标题
齐次接触黎曼三流形
DOI：
10.1215/ijm/1256045043
复制
发表时间：
1998
影响因子：
0.6
通讯作者：
D. Perrone
中科院分区：
文献类型：
--
作者：
D. Perrone
一个接触流形(M，O9)称为齐次流形[8]，如果有一个连通李群G传递地作用于M上的一组微分同胚使接触形式保持不变。众所周知，这一类推广了奇维球面给出的一类接触流形。如果g是伴随于co的度量，且G是传递地表现为保持co不变的等距群，则(co，g)称为M上的齐次接触黎曼结构。当(M，co)是紧致齐次接触流形时，通过Boothby-Wang纤维，人们可以考虑M上的齐次Sasakian结构(co，g)。在本文中，Goldberg 10]证明了球面是唯一单连通的齐次接触流形，它可以配备正截面曲率的不变接触度量(我们注意到齐次黎曼流形是完备的，因此当其截面曲率为正时是紧的)。最近，文献[13]、[14]证明了球面S3、S和Stiefel流形TI(S3)是唯一具有齐次接触结构的n维紧致单连通流形。本文的目的是研究不带紧性条件的单连通齐次接触黎曼3-流形。在第三节中，我们证明了所有这些流形都是具有左不变接触黎曼结构的李群。在单模情形下，挠率r满足p0，其中6是Berger-Ebin算子[1]，p-Rp，即所谓的p-挠率。此外，由Chern-Hamilton[9]引入的Webster标量曲率W和挠率不变量Ilvll刻画了这种流形。特别地，3-球面S是唯一允许齐次接触黎曼结构的单连通3-流形，具有标量曲率r>；-2(1ilrll)2；此外，海森堡群H和李群SL(2，R)是唯一允许具有Webster标量曲率W0的单模齐次接触黎曼结构的3-单连通流形。最后，在第四节，我们证明了单模齐次接触黎曼3-流形是局部p对称的。
A contact manifold (M, o9) is said to be homogeneous [8] if there is a connected Lie group G acting transitively as a group of diffeomorphisms on M which leave the contact form co invariant. As is well known, this class extends the class of contact manifolds given by odd-dimensional spheres. If g is a metric associated to co and G is a group acting transitively as a group of isometries which leave co invariant, then (co, g) is called a homogeneous contact Riemannian structure on M. When (M, co) is a compact homogeneous contact manifold, by the Boothby-Wang fibration one can consider a homogeneous Sasakian structures (co, g) on M. In this context Goldberg 10] showed that the sphere is the only simply connected homogeneous contact manifold which can be equipped with an invariant contact metric of positive sectional curvature (we note that a homogeneous Riemannian manifold is complete and hence compact when its sectional curvatures are positive). More recently, it has been proved in [13],[14] that the spheres S3, S and the Stiefel manifold TI(s3) are the only compact simply connected n-dimensional manifolds, n 3, 5, which admit a homogeneous contact structure. The purpose of this paper is to study simply connected homogeneous contact Riemannian 3-manifolds without the condition of compactness. In Section 3, we prove that all these manifolds are Lie groups equipped with a left invariant contact Riemannian structure. In the unimodular case the torsion r satisfies p 0, where 6 is the Berger-Ebin operator [1] and p -rp, the so-called p-torsion. Moreover, the Webster scalar curvature W and the torsion invariant Ilvll, introduced by Chern-Hamilton [9], characterize such manifolds. In particular, the 3sphere S is the only simply connected 3-manifold which admits a homogeneous contact Riemannian structure, with scalar curvature r > -2(1 Ilrll )2, Moreover, the Heisenberg group H and the Lie group SL(2, R) are the only simply connected 3-manifolds which admit an unimodular homogeneous contact Riemannian structure with Webster scalar curvature W 0. Finally, in Section 4 we show that unimodular homogeneous contact Riemannian 3-manifolds are locally psymmetric.