Geometric mean of probability measures and geodesics of Fisher information metric

The space of all probability measures having positive density function on a connected compact smooth manifold $M$, denoted by $\mathcal{P}(M)$, carries the Fisher information metric $G$. We define geometric mean aid which we investigate geometry equipped with show that geodesic segment joining arbitrary $\mu_1$ and $\mu_2$ is expressed using normalized its endpoints. As an application, any two points $\mathcal{P}(M)$ can be joined unique geodesic. Moreover, prove $\ell$ defined $\ell(\mu_1, \mu_2):=2\arccos\int_M \sqrt{p_1\,p_2}\,d\lambda$, $\mu_i=p_i\,\lambda$, $i=1,2$ gives Riemannian distance $\mathcal{P}(M)$. It shown geodesics are minimal.