  A relation between gravity on Poisson manifolds proposed in arXiv:1508.05706
and Einstein gravity is investigated. The compatibility of the Poisson and
Riemann structures defines a unique connection, the contravariant Levi-Civita
connection, and leads to the idea of the contravariant gravity. The
Einstein-Hilbert-type action yields an equation of motion which is written in
terms of the analog of the Einstein tensor, and it includes couplings between
the metric and the Poisson tensor. The study of the Weyl transformation reveals
properties of those interactions. It is argued that this theory can have an
equivalent description as a system of Einstein gravity coupled to matter. As an
example, it is shown that the contravariant gravity on a two-dimensional
Poisson manifold can be described by a real scalar field coupled to the metric
in a specific manner.
