Research on Geometric invariant on Manifolds and Lie transformation groups

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(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.