Local invariants of divergence-free webs

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 geo...

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Domitrz, Wojciech, Zubilewicz, Marcin
Format: Article
Language:English
Subjects:
Canonical forms
Invariants
Isomorphism
Numerical relativity
Relativity
Webs
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2206.11295