Symmetries on compact pseudo-Hermitian manifolds
A pseudo-Hermitian manifold \((M^*,w)\) consists of a \(C^{\infty}\) manifold \(M^*\) of dimension \(2n+1\) and of a global real 1-form w for which there are locally 2n complex 1-forms \(w_ 1,...,w_ n\), \(w_{\bar 1},...,w_{\bar n}\) with \(w_{\bar i}=\bar w_ i\), that together with w determine a local basis for the cotangent space of \(M^*\) and such that \(dw=-i g_{\alpha {\bar \beta}}w_{\alpha}\wedge w_{{\bar \beta}},\) where \(g=(g_{\alpha {\bar \beta}})\) is a nondegenerate Hermitian matrix; if it is negative definite, then \((M^*,w)\) is said to be strongly pseudoconvex. In this paper the author proves the following theorem: Any compact, simply connected, pseudo- Hermitian strongly pseudoconvex, symmetric manifold \((M^*,w)\) is a real hypersurface of a holomorphic line bundle F over a symmetric Hermitian manifold, whose Kähler form is, up to a multiplicative constant, the Chern class of F. More specifically \((M^*,w)\) is an \(S^ 1\)-principal bundle associated to F. It would be interesting to know whether the condition that \((M^*,w)\) has to be strongly pseudoconvex may be dropped.
scientific article; zbMATH DE number 4063968