Tohoku Mathematical Journal
2005
June
SECOND SERIES VOL. 57, NO. 2

Tohoku Math. J.
57 (2005), 201-221

Title GEOMETRIC FLOW ON COMPACT LOCALLY CONFORMALLY KÄHLER MANIFOLDS

Author Yoshinobu Kamishima and Liviu Ornea

(Received July 7, 2003, revised March 12, 2004)
Abstract. We study two kinds of transformation groups of a compact locally conformally Kähler (l.c.K.) manifold. First, we study compact l.c.K. manifolds by means of the existence of holomorphic l.c.K. flow (i.e., a conformal, holomorphic flow with respect to the Hermitian metric.) We characterize the structure of the compact l.c.K. manifolds with parallel Lee form. Next, we introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms preserving the specific $G$-structure of l.c.K. manifolds. We show that compact l.c.K. manifolds with parallel Lee form admitting a non-compact holomorphic flow of LCR transformations are rigid: such a manifold is holomorphically isometric to a Hopf manifold with parallel Lee form.

2000 Mathematics Subject Classification. Primary 57S25; Secondary 53C55.
Key words and phrases. Locally conformally Kähler manifold, Lee form, contact structure, strongly pseudoconvex CR-structure, $G$-structure, holomorphic complex torus action, transformation groups.

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