Geodesics of Fisher-Rao metric on the open interior of the finite-dimensional simplex. I am curious about the explicit form of the geodesics of the Fisher-Rao metric tensor on the open interior of the n-dimensional simplex.
In the 2-dimensional case (only 1 parameter on the 2-simplex), it is easy to explicitely compute them, however, starting from the 3-dimensional case, the situation becomes very complicated.
Since I am not an expert in this field, I am not able to see if the complexity of the situation depends on my ignorance, or it is intrinsic of the problem.
Hence, I ask you if there are some general results that are known, for instance, some particular explicit form, the qualitative behaviour, the completeness, and so on.
Thank You 
 A: Actually, I came up with a relatively easy solution.
There is a diffeomorphism $f$ between the open interior of the simplex $\Delta_{0}=\{\vec{p}\in\mathbb{R}^{n}\,\colon\:p^{j}>0\:\forall j=1,...,n\,,\:\:\sum_{j}\,p^j = 1\}$ and the intersection $S^{n}_{+}$ of the unit sphere $S^{n}$ in $\mathbb{R}^n$ with the open positive orthant $\mathbb{R}^{n}_{+}=\{\vec{v}\in\mathbb{R}^{n}\,\colon v^{j}>0\:\forall j=1,...,n\}$ given by $\vec{v}=f(\vec{p})=(\sqrt{p^{1}},...,\sqrt{p^{n}})$.
Then, denoting with $i_{n}$ the canonical (smooth) immersion of $S^{n}$ into $\mathbb{R}^{n}$, it is a nice and easy exercise to check that the Fisher-Rao metric on $\Delta_{0}$ is precisely the pullback through $i\circ f$ of the Euclidean metric on $\mathbb{R}^{n}$, or, equivalently, it is the pullback through $f$ of the restriction to $S^{n}_{+}$ of the round metric on $S^{n}$.
Consequently, a geodesic of the Fisher-Rao metric on $\Delta_{0}^{n}$ is the inverse image through $f^{-1}$ of a great circle starting in $S^{n}_{+}$.
From the fact that a great circle starting in $S^{n}_{+}$ will exit $S^{n}_{+}$ we conclude that every geodesic of the Fisher-Rao metric tensor is not complete.
