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Symmetries of supergeometries related to nonholonomic superdistributions

Boris Kruglikov, Andrea Santi, Dennis The

Abstract

We extend Tanaka theory to the context of supergeometry and obtain an upper bound on the supersymmetry dimension of geometric structures related to strongly regular bracket-generating distributions on supermanifolds and their structure reductions.

Symmetries of supergeometries related to nonholonomic superdistributions

Abstract

We extend Tanaka theory to the context of supergeometry and obtain an upper bound on the supersymmetry dimension of geometric structures related to strongly regular bracket-generating distributions on supermanifolds and their structure reductions.

Paper Structure

This paper contains 40 sections, 34 theorems, 105 equations, 2 tables.

Table of Contents

  1. Introduction and the main results
  2. Bundles on supermanifolds and geometric-algebraic correspondence
  3. Supermanifolds
  4. Lie supergroups and their actions
  5. Fiber bundles and sections
  6. Vector bundles on supermanifolds
  7. Principal bundles on supermanifolds
  8. Subbundles
  9. Prolongation Structures
  10. Frame bundles
  11. $G$-structures on supermanifolds
  12. Superdistributions and algebraic prolongations
  13. Filtered geometric structures
  14. Geometric Prolongation
  15. First prolongation
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $\mathfrak{s}$ be the symmetry superalgebra of a bracket-generating, strongly regular filtered $G_0$-structure $(M,\mathcal{D},q)$, with Tanaka--Weisfeiler prolongation $\mathfrak{g}=\mathop{\rm pr}\nolimits(\mathfrak{m},\mathfrak{g}_0)$ of $(\mathfrak{m},\mathfrak{g}_0)$. Assume the reduced man

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 69 more