Torsion of $\alpha$-connections on the density manifold

We study the torsion of the $\alpha$-connections defined on the density manifold in terms of a regular Riemannian metric. In the case of the Fisher-Rao metric our results confirm the fact that all $\alpha$-connections are torsion free. For the $\alpha$-connections obtained by the Otto metric, we show that, except for $\alpha =-1$, they are not torsion free.