Tohoku Mathematical Journal

Geometric flow on compact locally conformally Kähler manifolds

Yoshinobu Kamishima and Liviu Ornea

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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

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

First available in Project Euclid: 27 June 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57S25: Groups acting on specific manifolds
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

Locally conformally Kähler manifold Lee form contact structure strongly pseudoconvex CR-structure G-structure holomorphic complex torus action transformation groups


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.

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