  Inspired by the Poisson Sigma Model and its relation to 2d gravity, we
consider models governing morphisms from TSigma to any Lie algebroid E, where
Sigma is regarded as d-dimensional spacetime manifold. We address the question
of minimal conditions to be placed on a bilinear expression in the 1-form
fields, S^ij(X) A_i A_j, so as to permit an interpretation as a metric on
Sigma. This becomes a simple compatibility condition of the E-tensor S with the
chosen Lie algebroid structure on E. For the standard Lie algebroid E=TM the
additional structure is identified with a Riemannian foliation of M, in the
Poisson case E=T^*M with a sub-Riemannian structure which is Poisson invariant
with respect to its annihilator bundle. (For integrable image of S, this means
that the induced Riemannian leaves should be invariant with respect to all
Hamiltonian vector fields of functions which are locally constant on this
foliation). This provides a huge class of new gravity models in d dimensions,
embedding known 2d and 3d models as particular examples.
