Invariants On the Geometric Manifolds with Group Actions

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We have studied a geometric structure on a (4n+3)-dimensional smooth manifold M which is an integrable, nondegenerate codimension 3 subbundle D on M whose fiber supports the structure of 4n-dimensional quaternionic vector space. We call it a psesuo-conformal quterninonic structure. This structure has a refinement which is said to be a psesuo-conformal quterninonic CR structure. The structure is thought of as a generalization of the quaternionic CR structure. In order to study this geometric structure on M, we single out an sp(1)-valued 1-form ω locally on a neighborhood U of M such that Ker ω = D|U. We shall construct the invariants on the pair (M, ω) whose vanishing implies that M is uniformized with respect to a finite dimensional flat quaternionic CR geometry. In fact we have proved the standard psesuo-conformal quterninonic structure on the spahere S^<4n+3> coincides with the standard pseudo-quaternionic CR structure on S^<4n+3> The invariants obtained on a (4n+3)-manifold M have the same formula as the curvature tensor of quaternionic (indefinite) Kaehler manifolds. From this viewpoint, we shall exhibit a quaternionic analogue of Chern-Moser's CR structure. As to the global existence of the 1-form ω on a (4n+3)-manifold M is related to the Pontrjagin classes. We have shown the relation that 2p_1(M)=(n+2)p_1(L). In particular, if 2p_1(M)=0, then there exists a global 1-form co on M which represents a pseudo-conformal quaternionic structure D. As a consequence, there exists a hyperoomplex structure {I, J, K} on D.