Frölicher-Nijenhuis bracket on manifolds with special holonomy

Abstract

In this article, we summarize our recent results on the study of manifolds with special holonomy via the Frölicher-Nijenhuis bracket. This bracket enables us to define the Frölicher-Nijenhuis cohomologies which are analogues of the $d^c$ and the Dolbeault cohomologies in Kähler geometry, and assigns an $L_{\infty }$-algebra to each associative submanifold. We provide several concrete computations of the Frölicher-Nijenhuis cohomology.

14 pages, To appear in a forthcoming volume of the Fields Institute Communications, entitled "Lectures and Surveys on G2 manifolds and related topics". arXiv admin note: text overlap with arXiv:1703.05133