Classification of Lagrangian Fibrations over a Klein Bottle

TL;DR

This paper completes the classification of regular Lagrangian fibrations over compact surfaces by classifying those over the Klein bottle, extending previous work on tori using integral affine structures.

Contribution

It provides a complete classification of regular Lagrangian fibrations over the Klein bottle, filling a gap in the understanding of such fibrations over non-orientable surfaces.

Findings

Classification of regular Lagrangian fibrations over the Klein bottle.

Extension of previous classifications from tori to the Klein bottle.

Use of integral affine structures to achieve the classification.

Abstract

This paper completes the classification of regular Lagrangian fibratiopns over compact surfaces. \cite{misha} classifies regular Lagrangian fibrations over $\mathbb{T}^2$. The main theorem in \cite{hirsch} is used in order to classify integral affine structures on the Klein bottle $K^2$ and, hence, regular Lagrangian fibrations over this space.