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.

arXiv comment22 pages, comments welcome

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