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arXiv · 2608.21513

On the Integrability of Distributions in $\mathbb{Z}$-graded Geometry

Rudolf Šmolka·Jan Vysoký
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Abstract

Integrability of distributions on $\mathbb{Z}$-graded manifolds is examined. First, the Local Frobenius theorem -- the local existence of flat coordinates for an involutive distribution -- is proved. Then, two different notions of an integral submanifold are discussed. It is found that, for the stronger of the two notions, the Global Frobenius theorem holds, but as one implication only: involutivity implies integrability. It is then shown that integrability implies a certain weaker version of involutivity. Simple counterexamples of the converse implications are given.

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Rudolf Šmolka, Jan Vysoký. 2026-08-21. On the Integrability of Distributions in $\mathbb{Z}$-graded Geometry. https://arxiv.org/abs/2608.21513

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