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Real involutive systems on compact Lie groups

Gabriel Araújo, Igor A. Ferra, Max R. Jahnke, Luis F. Ragognette

Abstract

On a compact connected Lie group $G$, we study the global solvability and the cohomology spaces of the differential complex associated with an essentially real involutive structure that is invariant under left translations. We prove that solvability in the first degree of the complex implies solvability in all other degrees, and furnish a converse for this fact under a certain commutativity hypothesis (that always holds when $G$ is a torus). Additionally, it is proved that the solvability holds when the structure comes from the Lie algebra of a closed subgroup of $G$. We also investigate real tube structures when $G$ is the base manifold.

Real involutive systems on compact Lie groups

Abstract

On a compact connected Lie group , we study the global solvability and the cohomology spaces of the differential complex associated with an essentially real involutive structure that is invariant under left translations. We prove that solvability in the first degree of the complex implies solvability in all other degrees, and furnish a converse for this fact under a certain commutativity hypothesis (that always holds when is a torus). Additionally, it is proved that the solvability holds when the structure comes from the Lie algebra of a closed subgroup of . We also investigate real tube structures when is the base manifold.
Paper Structure (12 sections, 21 theorems, 162 equations)

This paper contains 12 sections, 21 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convenient metrics, frames and their sub-Laplacians
  4. Decompositions of forms
  5. Decompositions in cohomology
  6. Globally solvable structures
  7. Structures whose integral subgroup is closed
  8. Tube structures and related equations
  9. Normalization
  10. A formula for cohomology
  11. Perturbations of the de Rham complex
  12. Proofs of the algebraic lemmas

Key Result

Theorem 1

If then for all $1 \leq q \leq n \doteq \mathop{\mathrm{\mathrm{rank}}}\nolimits (\mathcal{V})$. Moreover, in that case, we have topologically, where:

Theorems & Definitions (39)

  • Theorem 1
  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 4.1
  • proof
  • ...and 29 more