Local invariants of divergence-free webs

TL;DR

This paper introduces two local invariants for divergence-free webs, linking curvature and holonomy, to classify their triviality and provide canonical forms, with potential applications in numerical relativity.

Contribution

It defines new local invariants for divergence-free webs and establishes their role in classification and canonical form determination.

Findings

Triviality of invariants characterizes trivial divergence-free web-germs.

Provides a canonical form and invariants for generic planar divergence-free webs.

Discusses potential applications in numerical relativity.

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 S. Tabachnikov (1992), and a geometric one, inspired by the classical notion of planar 3-web holonomy defined by W. Blaschke and G. Thomsen (1928). 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 S. 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.