Integrating Nijenhuis Structures

Abstract

A Nijenhuis operator on a manifold $M$ is a $(1,1)$ tensor $\mathcal N$ whose Nijenhuis-torsion vanishes. A Nijenhuis operator $\mathcal N$ on $M$ determines a Lie algebroid structure $(TM)_{\mathcal N}$ on the tangent bundle $TM$. In this sense a Nijenhuis operator can be seen as an infinitesimal object. In this paper, we identify its "global counterpart". Namely, we show that when the Lie algebroid $(TM)_{\mathcal N}$ is integrable, then it integrates to a Lie groupoid equipped with appropriate additional structure responsible for $\mathcal N$, and viceversa, the Lie algebroid of a Lie groupoid equipped with such additional structure is of the type $(TM)_{\mathcal N}$ for some Nijenhuis operator $\mathcal N$. We illustrate our integration result in various examples.