  We formulate the fractional Ricci flow theory for (pseudo) Riemannian
geometries enabled with nonholonomic distributions defining fractional
integro-differential structures, for non-integer dimensions. There are
constructed fractional analogs of Perelman's functionals and derived the
corresponding fractional evolution (Hamilton's) equations. We apply in
fractional calculus the nonlinear connection formalism originally elaborated in
Finsler geometry and generalizations and recently applied to classical and
quantum gravity theories. There are also analyzed the fractional operators for
the entropy and fundamental thermodynamic values.
