## Tohoku Mathematical Journal

### Geometric flow on compact locally conformally Kähler manifolds

#### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 57, Number 2 (2005), 201-221.

Dates
First available in Project Euclid: 27 June 2005

https://projecteuclid.org/euclid.tmj/1119888335

Digital Object Identifier
doi:10.2748/tmj/1119888335

Mathematical Reviews number (MathSciNet)
MR2137466

Zentralblatt MATH identifier
1083.53068

Subjects
Primary: 57S25: Groups acting on specific manifolds

#### Citation

Kamishima, Yoshinobu; Ornea, Liviu. Geometric flow on compact locally conformally Kähler manifolds. Tohoku Math. J. (2) 57 (2005), no. 2, 201--221. doi:10.2748/tmj/1119888335. https://projecteuclid.org/euclid.tmj/1119888335

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