Geodesic completeness of the affine connection on compact integral-integral affine bases

Determine whether the torsion-free flat affine connection induced by an integral-integral affine structure on a compact manifold B is geodesically complete, a question that arises for bases of prequantized Lagrangian torus fibrations and constitutes a special case of the Markus conjecture asserting geodesic completeness under unimodular transition maps.

Background

In the paper, the base B of a regular Lagrangian torus fibration inherits an integral-integral affine structure from enhanced Arnol'd–Liouville charts. Such structures determine a torsion-free flat affine connection on B via the affine atlas.

Despite compactness of B in the contexts considered, the authors explicitly note that it is unknown whether the induced affine connection is geodesically complete. This uncertainty is linked to the Markus conjecture, which predicts geodesic completeness for closed affine manifolds whose transition maps have determinant ±1 (a condition satisfied by integral-integral affine atlases).