Gravity from Lie algebroid morphisms (Q701713)

The author raises the question of finding the conditions under which it is possible to define a reasonable theory of gravity on a two-dimensional manifold. Let \(M\) be a manifold and let \(S\) be a two-dimensional manifold. After recalling the definition of Lie algebroids over \(M\), the author determines minimal conditions to be placed in order to permit an interpretation as a metric on the surface \(S\). The main result is then proved and applied to particular Lie algebroids (the standard Lie algebroid : \(TM\) and the Poisson case). Finally, the author gives a detailled outlook with open questions and interesting ideas to be developed.