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