|Speaker :||Professor Yoshinobu Kamishima (Josai University)|
|Title :||HOMOGENEOUS SASAKI MANIFOLD G/H OF UNIMODULAR LIE GROUP G|
|Time :||2017-02-17 (Fri) 11:00 - 12:00|
|Place :||Seminar Room 617, Institute of Mathematics (NTU Campus)|
A pseudo-Hermitian structure on a smooth manifold consists of a pair where is a contact form, together with a complex structure defined on a contact subbundle ker . When the Reeb field for generates a 1-parameter group of pseudo-Hermitian transformations of , i.e. holomorphic transformations on ker , we call a standard pseudo-Hermitian manifold, equivalently stated as a Sasaki manifold. In addition if acts properly and freely, then is said to be a regular Sasaki manifold. Suppose that is a simply connected homogeneous Sasaki manifold. Then it follows that is regular such that .
Theorem. Let be a simply connected unimodular Lie group and a connected compact subgroup. Then a homogeneous Sasaki manifold fibers over a homogeneous Kahler manifold of reductive Lie group with fiber :
Moreover, we shall apply this theorem to determine the unimodular Sasaki groups. Essentially the above theorem gives a final part of a clas-sification to homogeneous Vaisman manifolds. In fact, locally conformal Kahler structure on a complex manifold is a -invariant 2-form satisfying for some closed 1-form . When the Lee field dual to is holomorphic Killing with respect to the Hermitian metric is said to be a Vaisman manifold. We have shown that any simply connected homogeneous Vaisman manifold is the product of with a regular Sasaki manifold up to modification. This is based on our work on homogeneous locally conformal Kahler manifolds with D.A. Alekseevsky, V. Cortes, K. Hasegawa and myself.