Topological method in Differential Geometry and Conformal theory

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Throughout the academic year 1997 to 1998, we have studied a geometric structure on complex manifolds which fits into the framework of conformal geometry.Review 1. S.Bochner introduced the basic curvature tensor on Kahier manifolds, called Bochner curvature tensor. This tensor can be also derived from the Weyl's principle on unitary group representation theory. When Hermitian manifolds are taken into account in the framework of conformal geometry, Tricerri and Vanhecke have defined a generalized Bochner curvature tensor on Hermitian manifolds. This curvature tensor coincides with the original Bochner curvature tensor when a Hermitian manifold is Kahler. They observed that the generalized Bochner curvature tensor is a conformal invariant on Hermitian metrics on a Hermitian manifold. Here the complex structure is fixed. As a consequence, when a Hermitian manifold is a locally conformal KThler manifold, the (generalized) Bochner curvature tensor has the same formula as the (original) Boch … More ner curvature tensor.Several years ago, we have classified compact Kahler manifolds with vanishing Bochner tensor. A Bochner curvature flat Hermitian manifold is defined to be a Hermitian manifold with vanishing (generalized) Bochner curvature tensor. Bochner curvature flat Hermitian geometry is Hermitian geometry whose metrics are Bochner curvature flat. It is a problem what kind of Hermitian geometry admits a Bochner curvature flat Hermitian geometry. It is unknown which compact Hermitian manifold supports a Bochner curvature fiat metric. In addition, the classification of compact Bochner curvature flat Hermitian manifolds is far from valid as well as the classification of compact conformally flat manifolds. We restrict our attension to the class of locally conformal Kahler manifolds. Then we have arrived at the following classification theorem, which is different from that of the Kahler manifolds.Theorem A.Let (M, g, J) be a 2n-dimensional compact locally conformal Kahler manifold (n>1). If the Bochner curvature tensor vanishes, then Al is conformally equivalent to one of the following locally conformal Kahler manifolds :(1) The complex projective space CP^n.(2) A complex euclidean space form T^n_/F (F*U(n)).(3) A complex hyperbolic space form H^n_/GAMMA (GAMMA*PU (n, 1)).(4) A fiber space H^m_*__<GAMMA>CP^<n-m>(GAMMA*PU(m, 1)*PU(n-m+1), m=1, 2, ・, n-1)(5) A Hopf manifold S^<2-1>*__<GAMMA>S^1(F*U(u)*S^1).(i) F is a finite group and GAMMA is a discrete cocompact subgroup which acts freely and properly discontinuously.(ii) The manifolds of the above (1), (2), (3), (4) are Kahler manifolds. When g is a Kahler metric from the beginning, (M, g, J) is holomorphically isometric (up to a constant scalar multip1e of the Kahler metric) to one of the above Kahler manifolds of (1), (2), (3), (4).Review 2. This part concerns a geometric structure on (4n+3)-dimensional smooth manifolds. The isometry group of quaternionic hyperbolic space acts transitively on the boundary sphere as projective transformations. It gives a geometry (PSp(n+1, 1), S^<4n+3>). A(4n+3)-manifold locally modelled on this geometry is said to be a spherical pseudo-quaternionic manifold. We discuss a Carnot-Caratheodory structure on spherical pseudo-quaternionic manifolds in connection with the Sasakian 3-structure. Using superrigidity in quaternionic hyperbolic group, we have proved the geometric rigidity of compact spherical pseudo-quaternionic (4n+3)-manifolds when the fundamental group is isomorphic to either an amenable group or a quaternionic hyperbolic group. A spherical pseudo-quaternionic structure is a geometric structure on a (4n+3)-manifold locally modelled on the sphere S^<4n+3> with coordinate changes lying in the Lorentz group PSp(n+1, 1). Here PSp(n+1, 1) is isomorphic to the isometry group Iso(H^<n+1>_) of the quaternionic hyperbolic space H^<n+l>_. The space H^<n+1>_ has the projective compactification whose boundary is the sphere S^<4n+3> on which PSp(n+1, 1) acts as projective transformations. The pair (PSp(n+1, 1), S^<4n+3>) is said to be spherical pseudo-quaternionic geometry. A (4n+3)-manifold locally modelled on this geometry is said to be a spherical pseudo-quaterniortic manifold. By using the Margulis' superrigidity by Corlette, we proved that the following rigidity.Theorem B.Let M be a compact spherical pseudo-quaternionic (4n+3)-manifold whose fundamental group pi(M) is isomorphic to a discrete uniform subgroup of PSp(m, 1) for some m where 2*m*n. Then M is pseudo-quaternionically isomorphic to the double coset spaceSp(m)*DELTASp(l)*Sp(n-m)*Sp(m, 1)・Sp(n-m+1)/GAMMA where m=2, ・, n. Less