Speaker : Professor Yoshinobu Kamishima (Josai University) HOMOGENEOUS SASAKI MANIFOLD G/H OF UNIMODULAR LIE GROUP G 2017-02-17 (Fri) 11:00 - 12:00 Seminar Room 617, Institute of Mathematics (NTU Campus) A pseudo-Hermitian structure on a smooth manifold $M$ consists of a pair $(\omega, J )$ where $\omega$ is a contact form, together with a complex structure $J$ defined on a contact subbundle ker $\omega$. When the Reeb field $A$ for $\omega$ generates a 1-parameter group $T$ of pseudo-Hermitian transformations of $M$, i.e. holomorphic transformations on ker $\omega$, we call $(M, \omega, J, A)$ a standard pseudo-Hermitian manifold, equivalently stated as a Sasaki manifold. In addition if $T$ acts properly and freely, then $M$ is said to be a regular Sasaki manifold. Suppose that $(G/H, \omega, J)$ is a simply connected homogeneous Sasaki manifold. Then it follows that $G/H$ is regular such that $T = S^{1} or \mathbb{R}$. Theorem. Let $G$ be a simply connected unimodular Lie group and $H$ a connected compact subgroup. Then a homogeneous Sasaki manifold $G/H$ fibers over a homogeneous Kahler manifold $PG/PH$ of reductive Lie group $PG$ with fiber $T$ : $T \rightarrow G/H \overset{P}{\rightarrow} PG/PH$. 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 $X$ is a $J$ -invariant 2-form $\Omega$ satisfying $d\Omega = \theta \wedge \Omega$ for some closed 1-form $\theta$. When the Lee field $\xi$ dual to $\theta$ is holomorphic Killing with respect to the Hermitian metric $g = \Omega \circ J, X$ is said to be a Vaisman manifold. We have shown that any simply connected homogeneous Vaisman manifold is the product of $\mathbb{R}$ 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.