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https://doi.org/10.2139/ssrn.5...
Article . 2025 . Peer-reviewed
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Homogeneity Supermanifolds and Homogeneous Darboux Theorem

Authors: Grabowska, Katarzyna; Grabowski, Janusz;

Homogeneity Supermanifolds and Homogeneous Darboux Theorem

Abstract

We develop the concept of a homogeneity supermanifold, which is more general than most of the approaches to graded manifolds present in the literature. Instead of doing it via ringed spaces and sheaves of graded algebras, we prefer a purely differential geometric language and view gradings as provided by an additional structure on supermanifolds, namely the weight vector field. This gives us a privileged atlas with homogeneous local coordinates. The fact that the transition maps `preserve weights' is then automatic, but weights of homogeneous coordinates may vary from one chart to the other. Moreover, such local homogeneous coordinates may have arbitrary real weights, not only integers. We study homogeneity submanifolds, homogeneity Lie supergroups, tangent and cotangent lifts of homogeneity structures, homogeneous distributions and codistributions, and other related concepts. The main achievements in this framework are proofs of the homogeneous Poincar\'e Lemma and a homogeneous analog of the Darboux Theorem.

Comment: 39 pages

Keywords

Mathematics - Differential Geometry, Mathematics - Symplectic Geometry, 53D05, 53D10, 53B99, 57N16, 58A50 (Primary), 14A22, 17A30, 58A05, 58A50 (Secondary), Mathematical Physics

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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
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