Local invariants of divergence-free webs

Abstract

The objects of our study are webs in the geometry of volume-preserving diffeomorphisms. We introduce two local invariants of divergence-free webs: a differential one, directly related to the curvature of the natural connection of a divergence-free 2-web introduced by Tabachnikov (Diff Geom Appl 3:265-284, 1993), and a geometric one, inspired by the classical notion of planar 3-web holonomy defined by Blaschke and Bol (Geometrie der Gewebe. Grundlehren der mathematischen Wissenschaften, vol. 49. Springer, Berlin, 1938). We show that triviality of either of these invariants characterizes trivial divergence-free web-germs up to equivalence. We also establish some preliminary results regarding the full classification problem, which jointly generalize the theorem of Tabachnikov on normal forms of divergence-free 2-webs. They are used to provide a canonical form and a complete set of invariants of a generic divergence-free web in the planar case. Lastly, the relevance of local triviality conditions and their potential applications in numerical relativity are discussed.