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Locally conformally Kähler metrics on Hopf surfaces

Paul Gauduchon, Liviu Ornea (1998)

Annales de l'institut Fourier

A primary Hopf surface is a compact complex surface with universal cover ℂ 2 - { ( 0 , 0 ) } and cyclic fundamental group generated by the transformation ( u , v ) ↦ ( α u + λ v m , β v ) , m ∈ ℤ , and α , β , λ ∈ ℂ such that ∣ α ∣ ≥ ∣ β ∣ > 1 and ( α - β m ) λ = 0 . Being diffeomorphic with S 3 × S 1 Hopf surfaces cannot admit any Kähler metric. However, it was known that for λ = 0 and ∣ α ∣ = ∣ β ∣ they admit a locally conformally Kähler metric with parallel Lee form. We here provide the construction of a locally conformally Kähler metric with parallel Lee form for all primary Hopf surfaces of class 1 ( λ = 0 ). We also show...

Locally symmetric immersions

José Carmelo González-Dávila, Lieven Vanhecke (1999)

Czechoslovak Mathematical Journal

We use reflections with respect to submanifolds and related geometric results to develop, inspired by the work of Ferus and other authors, in a unified way a local theory of extrinsic symmetric immersions and submanifolds in a general analytic Riemannian manifold and in locally symmetric spaces. In particular we treat the case of real and complex space forms and study additional relations with holomorphic and symplectic reflections when the ambient space is almost Hermitian. The global case is also...

Logarithmic Surfaces and Hyperbolicity

Gerd Dethloff, Steven S.-Y. Lu (2007)

Annales de l’institut Fourier

In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate.In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2 , any...

Lorentzian geometry in the large

John Beem (1997)

Banach Center Publications

Lorentzian geometry in the large has certain similarities and certain fundamental differences from Riemannian geometry in the large. The Morse index theory for timelike geodesics is quite similar to the corresponding theory for Riemannian manifolds. However, results on completeness for Lorentzian manifolds are quite different from the corresponding results for positive definite manifolds. A generalization of global hyperbolicity known as pseudoconvexity is described. It has important implications...

Lorentzian similarity manifolds

Yoshinobu Kamishima (2012)

Open Mathematics

An (m+2)-dimensional Lorentzian similarity manifold M is an affine flat manifold locally modeled on (G,ℝm+2), where G = ℝm+2 ⋊ (O(m+1, 1)×ℝ+). M is also a conformally flat Lorentzian manifold because G is isomorphic to the stabilizer of the Lorentzian group PO(m+2, 2) of the Lorentz model S m+1,1. We discuss the properties of compact Lorentzian similarity manifolds using developing maps and holonomy representations.

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