Research on Geometric invariant on Manifolds and Lie transformation groups

(1)我们研究了4N+3-流形M上的一个可积、非退化余维3-子丛D，它的纤维支持4N维四元数向量空间的结构。它被认为是四元数CR结构的推广。我们在一个邻域U上挑出一个SP(1)值的1-形式ω，使得零ω=DIu，并构造了(M，ω)上的曲率不变量，它的消失使平坦的四元数CR几何一致.在M上得到的不变量与伪四元数Kaehler 4N流形具有相同的公式。(2)Long和Reid证明了n&gt；2维黎曼平坦流形的微分同胚类是作为完全有限体积实双曲流形的某一尖截面产生的。对于复双曲情形，D.B.McReynolds证明了每个三维次流形都微分同胚于一个完全有限体积复双曲2-orbor流形的一个尖点截面。我们利用Seifert纤维来研究这一实现问题。设π是n维晶学群。然后有一个忠实表示B：πZ^n×GL(n，Z)。特别地，每一个紧致黎曼平坦流形R^n/π都可以实现为一个完全有限体积实双曲流形的一个尖点截面。(3)证明了每一个紧致非球面齐次流形都是基上纤维上具有解几何的纤维的全空间，它是一个局部对称的非正曲率的流形。我们在这种流形上构造了一个迭代内射Seifert纤维结构，从而证明了这种流形之间的每个同伦等价都是由一个微分同胚导出的。特别地，两个紧致齐次非球面流形是微分同构的当且仅当它们的基本群同构。

(1) We have studied an integrable, nondegenerate codimension 3 -subbundle D on a 4n+3- manifold M whose fiber supports the structure of 4n-dimensional quaternionic vector space. It is thought of as a generalization of quaternionic CR structure. We single out an sp (1)-valued 1-form ω loally on a neighborhood U such that Null ω= DIU and construct the curvature invariant on (M,ω) whose vanishing gives a uniformization to flat quaternionic CR geometry. The invariant obtained on M has the same formula as that of pseudo-quaternionic Kaehler 4n-manifolds. From this viewpoint, we have exhibited a quaternionic analogue of Chern-Moser's CR structure.(2) Long and Reid have shown that the diffeomorphism class of every Riemannian flat manifold of dimension n>2 arises as some cusp cross-section of a complete finite volume real hyperbolic orbifold. For the complex hyperbolic case, D. B. McReynolds proved that every 3-dimensional infranilmanifold is diffeomorphic to a cusp cross-section of a complete finite volume complex hyperbolic 2-orbifold. We study this realization problem by using Seifert fibration. Let π be an n-dimensional crystallographic group. Then there is a faithful representation B: π Z^n×GL (n, Z). In particular, every compact Riemannian flat orbifold R^n/π can be realized as a cusp cross-section of a complete finite volume real hyperbolic orbifold.(3) We have proved that every compact aspherical homogeneous manifold is the total space of a fibration with solv-geometry on the fibers over a base which is a locally symmetric orbifold of non-positive curvature. We construct an iterated injective Seifert fibered structure on such fibrations, and this allows to prove that every homotopy equivalence between such manifolds is induced by a diffeomorphism. In particular, two compact homogeneous aspherical manifolds are diffeomorphic if and only if their fundamental groups are isomorphic.