Existence of Lie algebroids on the tangent bundle with a given anchor map of constant rank

Abstract

We show that given a constant rank linear map, $K:TM\to TM$, there exists a Lie algebroid with K as its anchor map, if and only if the image distribution, ImK, is involutive. As a byproduct, a new example of Lie algebroid bracket associated with a regular foliation is obtained through the projector onto the involutive distribution. The Lie algebroid bracket is not defined on the involutive distribution but on the whole space of vector fields of the manifold.