Bi-Lagrangian structures on nilmanifolds

TL;DR

This paper classifies 6-dimensional nilpotent Lie algebras that admit bi-Lagrangian structures, exploring their existence and curvature properties, thus advancing understanding of symplectic geometry on nilmanifolds.

Contribution

It identifies which 6-dimensional nilpotent Lie algebras support bi-Lagrangian structures and computes the curvature of their canonical connections, extending previous classifications.

Findings

16 out of 26 symplectic nilpotent Lie algebras admit bi-Lagrangian structures

10 nilpotent Lie algebras do not admit such structures

Curvature of the canonical connection is explicitly calculated

Abstract

We study bi-Lagrangian structures (a symplectic form with a pair of complementary Lagrangian foliations, also known as para-K\"ahler or K\"unneth structures) on nilmanifolds of dimension less than or equal to 6. In particular, building on previous work of several authors, we determine which 6-dimensional nilpotent Lie algebras admit a bi-Lagrangian structure. In dimension 6, there are (up to isomorphism) 26 nilpotent Lie algebras which admit a symplectic form, 16 of which admit a bi-Lagrangian structure and 10 of which do not. We also calculate the curvature of the canonical connection of these bi-Lagrangian structures.