  There were elaborated different models of Finsler geometry using the Cartan
(metric compatible), or Berwald and Chern (metric non-compatible) connections,
the Ricci flag curvature etc. In a series of works, we studied (non)commutative
metric compatible Finsler and nonholonomic generalizations of the Ricci flow
theory [see S. Vacaru, J. Math. Phys. 49 (2008) 043504; 50 (2009) 073503 and
references therein]. The goal of this work is to prove that there are some
models of Finsler gravity and geometric evolution theories with generalized
Perelman's functionals, and correspondingly derived nonholonomic Hamilton
evolution equations, when metric noncompatible Finsler connections are
involved. Following such an approach, we have to consider distortion tensors,
uniquely defined by the Finsler metric, from the Cartan and/or the canonical
metric compatible connections. We conclude that, in general, it is not possible
to elaborate self-consistent models of geometric evolution with arbitrary
Finsler metric noncompatible connections.
