  Motivated by the geometrical structures of quantum mechanics, we introduce an
almost-complex structure $J$ on the product $M\times M$ of any parallelizable
statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and
a metric-like tensor on $M\times M$ from a divergence function. These tensors
may be pulled back to $M$, and we compute them in the case of an N-dimensional
symplex with respect to the Kullback-Leibler relative entropy, and in the case
of (a suitable unfolding space of) the manifold of faithful density operators
with respect to the von Neumann-Umegaki relative entropy.
