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A Darboux classification of homogeneous Pfaffian forms on graded manifolds

Janusz Grabowski, Asier López-Gordón

TL;DR

The paper develops a Darboux-type local classification for homogeneous Pfaffian forms on (graded) supermanifolds by leveraging a constant-rank characteristic distribution $χ(\alpha)=\ker(\alpha)\cap\ker(\mathrm{d}\alpha)$ to define the class, and a weight vector field $\nabla$ to impose homogeneity. It proves a homogeneous Poincaré lemma and a homogeneous Frobenius theorem to obtain homogeneous Darboux coordinates, yielding canonical pictures for homogeneous presymplectic forms and homogeneous one-forms of constant class, and shows how standard Darboux, contact, and presymplectic results arise as special cases. The work provides concrete homogeneous normal forms, including explicit expressions for presymplectic and contact-type objects, and supplies a suite of examples and Pfaffian equations illustrating the reach and limitations of homogeneous Darboux coordinates in supergeometry. This unifies classical and graded-geometric Darboux theory and offers tools for local homogenized classifications on N-manifolds and vector superbundles with broad applications in graded differential geometry.

Abstract

We study the local classification problem for differential Pfaffian forms on a supermanifold $M$ that are homogeneous with respect to a given homogeneity structure on $M$. The best-known homogeneity structures are those associated with linearity on a vector bundle. The aim is to show that, for a homogeneous form of given degree, there are `Darboux coordinates' which are homogeneous. As a consequence, we obtain Darboux-type normal pictures for homogeneous forms, recovering the classical Darboux theorems, as well as their contact and presymplectic counterparts as special cases. To obtain an analog of Darboux's classification in the supergeometric context, we define a class of a differential form $α$ by the rank of the characteristic distribution $χ(α)$, being the intersection of kernels of $α$ and $dα$. We show that, under suitable regularity and constant-rank assumptions, this distribution completely controls the local equivalence problem for homogeneous Pfaffian forms. Our results hold as well for ordinary (purely even) manifolds.

A Darboux classification of homogeneous Pfaffian forms on graded manifolds

TL;DR

The paper develops a Darboux-type local classification for homogeneous Pfaffian forms on (graded) supermanifolds by leveraging a constant-rank characteristic distribution to define the class, and a weight vector field to impose homogeneity. It proves a homogeneous Poincaré lemma and a homogeneous Frobenius theorem to obtain homogeneous Darboux coordinates, yielding canonical pictures for homogeneous presymplectic forms and homogeneous one-forms of constant class, and shows how standard Darboux, contact, and presymplectic results arise as special cases. The work provides concrete homogeneous normal forms, including explicit expressions for presymplectic and contact-type objects, and supplies a suite of examples and Pfaffian equations illustrating the reach and limitations of homogeneous Darboux coordinates in supergeometry. This unifies classical and graded-geometric Darboux theory and offers tools for local homogenized classifications on N-manifolds and vector superbundles with broad applications in graded differential geometry.

Abstract

We study the local classification problem for differential Pfaffian forms on a supermanifold that are homogeneous with respect to a given homogeneity structure on . The best-known homogeneity structures are those associated with linearity on a vector bundle. The aim is to show that, for a homogeneous form of given degree, there are `Darboux coordinates' which are homogeneous. As a consequence, we obtain Darboux-type normal pictures for homogeneous forms, recovering the classical Darboux theorems, as well as their contact and presymplectic counterparts as special cases. To obtain an analog of Darboux's classification in the supergeometric context, we define a class of a differential form by the rank of the characteristic distribution , being the intersection of kernels of and . We show that, under suitable regularity and constant-rank assumptions, this distribution completely controls the local equivalence problem for homogeneous Pfaffian forms. Our results hold as well for ordinary (purely even) manifolds.
Paper Structure (11 sections, 15 theorems, 81 equations)

This paper contains 11 sections, 15 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Introduction to homogeneity supermanifolds
  3. Vector superbundles
  4. Darboux theorems for homogeneous forms
  5. Homogeneous presymplectic forms
  6. Homogeneous one-forms of constant class
  7. Examples of homogeneous Pfaffian forms
  8. Standard Darboux theorems
  9. Non-trivial homogeneous forms of weight zero
  10. Exact symplectic forms
  11. Homogeneous Pfaffian equations

Key Result

Proposition 2.4

An even vector field $\nabla$ on a supermanifold $M$ is a weight vector field if and only if $\nabla(M)$ is locally linear, with $D_m\nabla$ diagonalizable, for each $m\in|M|$ in the zero-locus of $\nabla$. In particular, non-vanishing even vector fields are weight vector fields. If, in local homoge where $n$ is the total dimension of $M$, then:

Theorems & Definitions (34)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7: Homogeneity category
  • Example 3.1
  • Theorem 3.2: Lifts of homogeneity structures
  • Definition 3.3
  • ...and 24 more