Differential Geometry of Microlinear Frolicher Spaces III

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.