  We prove that the Riemannian geometry of almost K\"ahler manifolds can be
expressed in terms of the Poisson algebra of smooth functions on the manifold.
Subsequently, K\"ahler-Poisson algebras are introduced, and it is shown that a
corresponding purely algebraic theory of geometry and curvature can be
developed. As an illustration of the new concepts we give an algebraic proof of
the statement that a bound on the (algebraic) Ricci curvature induces a bound
on the eigenvalues of the (algebraic) Laplace operator, in analogy with the
well-known theorem in Riemannian geometry. As the correspondence between
Poisson brackets of smooth functions and commutators of operators lies at the
heart of quantization, a purely Poisson algebraic proof of, for instance, such
a "Gap Theorem", might lead to an understanding of spectral properties in a
corresponding quantum mechanical system.
