Research on Geometric invariant on Manifolds and Lie transformation groups

(1)本文研究了4 n +3-流形M上的可积的非退化余维3 -子丛D，其纤维支持4 n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。我们在邻域U上局部地选出一个sp（1）值1-形式ω，使得ω= DIU，并构造了（M，ω）上的曲率不变量，它的消失给出了平坦四元数CR几何的一致化. M上的不变量与伪四元数Kaehler 4 n-流形的不变量具有相同的公式。从这个观点出发，我们展示了一个四元数类似的陈-莫泽的CR结构。(2)Long和Reid已经证明了维数n>2的每个黎曼平坦流形的自同构类是由完备有限体积真实的双曲轨道的尖点截面产生的。对于复双曲情形，D. B。McReynolds证明了每一个三维infranilmanifold是一个完整的有限体积复双曲2-orbifold的尖点横截面的一个同态。我们使用塞弗特纤维化研究这个实现问题。设π是一个n维晶体群。然后有一个忠实的表示B：π Z^n×GL（n，Z）。特别地，每一个紧致的黎曼平坦轨道褶皱R^n/π都可以被实现为一个完整的有限体积真实的双曲轨道褶皱的尖点截面。(3)我们证明了每个紧致非球面齐次流形都是在一个非正曲率的局部对称轨道上的纤维上具有解几何的纤维化的全空间。我们构造了一个迭代内射塞弗特纤维结构，这样的纤维化，这使得可以证明，每一个同伦等价等流形是由一个同胚诱导。特别地，两个紧致齐次非球面流形是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.