Differential Geometry of Microlinear Frolicher Spaces III

TL;DR

This paper advances the differential geometry of microlinear Frolicher spaces by proving the Frolicher-Nijenhuis bracket satisfies the graded Jacobi identity and that Lie derivation preserves this bracket, emphasizing geometric perspectives.

Contribution

It establishes the graded Jacobi identity for the Frolicher-Nijenhuis bracket and shows Lie derivation preserves it within the context of microlinear Frolicher spaces.

Findings

Frolicher-Nijenhuis bracket satisfies graded Jacobi identity

Lie derivation preserves the Frolicher-Nijenhuis bracket

The approach is highly geometric compared to original algebraic definitions

Abstract

As the third of our series of papers on differential geometry of microlinear Frolicher spaces, this paper is devoted to the Frolicher-Nijenhuis calculus of their named bracket. The main result is that the Frolicher-Nijenhuis bracket satisfies the graded Jacobi identity. It is also shown that the Lie derivation preserves the Frolicher-Nijenhuis bracket. Our definitions and discussions are highly geometric, while Frolicher and Nijenhuis' original definitions and discussions were highly algebraic.