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

We show that given a constant rank linear map, K : TM & RARR;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. & COPY; 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons .org /licenses /by -nc -nd /4 .0/).

Automated summary

We prove that for a given constant rank linear map K: TM→TM, there exists a Lie algebroid with K as its anchor map if and only if the image distribution ImK is involutive. As a result, a new example of Lie algebroid bracket associated with a regular foliation is obtained by projecting onto the involutive distribution. The Lie algebroid bracket is defined not only on the involutive distribution but also on the whole space of vector fields of the manifold.