Geometric structure on geometric manifolds which admit Lie group transformations and various Rigidity

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We have examined the following projects.(1) Approach from geometry and topology concerning conformally Fefferman-Lorentz manifolds.(1) We proved the analogue of the Obata-Ferrand theorem to Lorentz manifolds.(2) We proved the Kuiper's problem that if the developing map is not surjective on the universal cover of S^<n-1,1>, then it is a covering map onto its image under the existence of causal vector fields.(2) We studied the infrasolv-fiber space structure and the smooth rigidity on the closed aspherical manifolds of the mixed type. As an application, we proved the smooth rigidity of compact aspherical homogeneous manifolds.