  Models of geometric flows pertaining to $\mathcal{R}^2$ scale invariant
(super) gravity theories coupled to conformally invariant matter fields are
investigated. Related to this work are supersymmetric scalar manifolds that are
isomorphic to the K\"{a}hlerian spaces $\mathcal{M}_n=SU(1,1+k)/U(1)\times
SU(1+k)$ as generalizations of the non-supersymmetric analogs with
$SO(1,1+k)/SO(1+k)$ manifolds. For curved superspaces with geometric evolution
of physical objects, a complete supersymmetric theory has to be elaborated on
nonholonomic (super) manifolds and bundles determined by non-integrable
superdistributions with additional constraints on (super) field dynamics and
geometric evolution equations. We also consider generalizations of Perelman's
functionals using such nonholonomic variables which result in the decoupling of
geometric flow equations and Ricci soliton equations with supergravity
modifications of the $R^2$ gravity theory. As such, it is possible to construct
exact non-homogeneous and locally anisotropic cosmological solutions for
various types of (super) gravity theories modelled as modified Ricci soliton
configurations. Such solutions are defined by employing the general ansatz
encompassing coefficients of generic off-diagonal metrics and generalized
connections that depend generically on all spacetime coordinates. We consider
nonholonomic constraints resulting in diagonal homogeneous configurations
encoding contributions from possible nonlinear parametric geometric evolution
scenarios, off-diagonal interactions and anisotropic polarization/ modification
of physical constants. We also analyze the conditions under which such modified
mimetic type theories and solutions reproduce the Starobinsky inflationary
models in the double scalar approach.
