Geometric Structures on Manifolds and Representations of Fundamental Group

2n + 1维光滑流形M上的CR-结构由一对（omega，J）组成，满足（i）omega是M的切触形式，（ii）设ω omega = {X * TM* omega（X）= 0}是切丛TM的余维1子丛.然后在ω上有一个复杂的结构J。也就是说，J是π ω上的几乎复结构，当π ω <cross product>C = T^&lt;1，0&gt; + T^&lt;0，1&gt;是正则分裂时，它遵循[T^&lt;1，0&gt;，T^&lt;1，0&gt;] * T^&lt;1，0&gt;。此外，如果<10>由Q（X，Y）= d-ω（X，JY）定义的Hermitian对Q：T^ XT^&lt;1，0&gt; -&gt; C是正定的，则CR-结构是严格伪凸的. 1958年Boothby和Wang引入了光滑流形上的正则切触结构，并建立了振动定理。规则CR结构将被定义为其底层接触结构是规则的CR结构。然后推广Boothby和Wang的结果：M^&lt;2n+1&gt;有正则CR-结构当且仅当M是Kahler流形N上的主圆丛pi：M -&gt; N，其基本2-形式Ω满足以下性质：（1）丛的e-cycle类由整上圈[Ω] * H^2（N ; Z）表示。(2)其中eta是M的联络形式。

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.