This paper has two purposes: the first is to study several structures on manifolds in the general setting of real and complex differential geometry; the second is to apply this study to Teichmüller theory. We primarily focus on bi-Lagrangian structures, which are the data of a symplectic structure and a pair of transverse Lagrangian foliations, and are equivalent to para-Kähler structures. First we carefully study real and complex bi-Lagrangian structures and discuss other closely related structures and their interrelationships. Next we prove the existence of a canonical complex bi-Lagrangian structure in the complexification of any real-analytic Kähler manifold and showcase its properties. We later use this bi-Lagrangian structure to construct a natural almost hyper-Hermitian structure. We then specialize our study to moduli spaces of geometric structures on closed surfaces, which tend to have a rich symplectic structure. We show that some of the recognized geometric features of these moduli spaces are formal consequences of the general theory, while revealing other new geometric features. We also gain clarity on several well-known results of Teichmüller theory by deriving them from pure differential geometric machinery.
SEEK ID: https://publications.h-its.org/publications/960
Research Groups: Groups and Geometry
Publication type: Misc
Journal: arXiv,math.DG,1708.09145
Citation: arXiv,math.DG,1708.09145
Date Published: 2017
URL: https://arxiv.org/abs/1708.09145
Registered Mode: imported from a bibtex file
Views: 6121
Created: 30th Jan 2020 at 09:39
Last updated: 5th Mar 2024 at 21:24
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