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 ($\mathrm{LCR}$) 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 $\mathrm{LCR}$ transformations are rigid: such a manifold is holomorphically isometric to a Hopf manifold with parallel Lee form.
Citation
Yoshinobu Kamishima. Liviu Ornea. "Geometric flow on compact locally conformally Kähler manifolds." Tohoku Math. J. (2) 57 (2) 201 - 221, 2005. https://doi.org/10.2748/tmj/1119888335
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