Geometric Structures on Manifolds and Representations of Fundamental Group

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A CR-structure on a 2n + 1 dimensional smooth manifold M consists of a pair (omega, J) satisfying that (i) omega is a contact form of M , and (ii) Let Null omega = {X * TM* omega (X) = 0} which is a codimension 1 sub-bundle of the tangent bundle TM. Then there is a complex structure J on Null omega. Namely J is an almost complex structure on Null omega and when Null omega <cross product> C = T^<1,0> + T^<0,1> is the canonical splitting, it follows that [T^<1,0>, T^<1,0>] * T^<1,0>. In addition a CR-structure is strictly pseudo-convex if the Hermitian pairing Q : T^<10>XT^<1,0> -> C defined by Q (X, Y) = d-omega (X, JY) is positive definite. In 1958 Boothby and Wang introduced the regular contact structure on smooth manifolds and they have established the vibration theorem. A regular CR-structure will be defined as a CR-structure whose underlying contact structure is regular. Then the Boothby and Wang's result will be generalized as follows : M^<2n+1> admits a regular CR-structure if and only if M is a principal circle bundle pi : M -> N over a Kahler manifold N whose fundamental 2-form OMEGA satisfies the following properties ; (1) the euler class of the bundle is represented by an integral cocycle [OMEGA] * H^2 (N ; Z). (2) deta = pi^*OMEGA where eta is a connection form of M.