Maurer-Cartan equation in the DGLA of graded derivations

Abstract

Let M be a smooth manifold and $\Phi$ a differential 1-form on M with values in the tangent bundle TM. We construct canonical solutions $e_{\Phi }$ of Maurer-Cartan equation in the DGLA of graded derivations D*(M) of differential forms on M by means of deformations of d depending on $\Phi$. This yields to a classification of the canonical solutions of the Maurer-Cartan equation according to their type: $e_{\Phi }$ is of finite type r if there exists $r\in N$ such that ${\Phi }^r[\Phi ,\Phi {]}_{FN}=0$ and r is minimal with this property, where $[.,.{]}_{FN}$ is the Frölicher-Nijenhuis bracket. A distribution $\xi \subset TM$ of codimension k > 1 is integrable if and only if the canonical solution $e_{\Phi }$ associated to the endomorphism $\Phi$ of TM which is trivial on $\xi$ and equal to the identity on a complement of $\xi$ in TM is of finite type $\le 1$, respectively of finite type 0 if k = 1.