LIE TENSOR PRODUCT MANIFOLDS

We study the geometric structures of parabolic geometries. A parabolic geometry is defined by a parabolic subgroup of a simple Lie group corresponding to a subset of the positive simple roots. We say that a parabolic geometry is fundamental if it is defined by a subset corresponding to a single simple root. In this paper we will be mainly concerned with such fundamental parabolic geometries. Fundamental geometries for the Lie algebra of A(n) type are Grassmann structures. For B-n, C-n, D-n types, we investigate the geometric feature of the fundamental geometries modeled after the quotients of the real simple groups of split type by the parabolic subgroups. We name such geometries Lie tensor product structures. Especially, we call Lie tensor metric structure for B-n or D-n type and Lie tensor symplectic structure for C-n type. For each manifold with a Lie tensor product structure, we give a unique normal Cartan connection by the method due to Tanaka. Invariants of the structure are the curvatures of the connection.