arXiv:1606.06884v3 [hep-th] 28 Jun 2017 On Supersymmetric Geometric Flows and R 2 Inflation From Scale Invariant Supergravity Subhash Rajpoot California State University at Long Beach, Long Beach, California, USA email: Subhash.Rajpoot@csulb.edu Sergiu I. Vacaru ∗ Physics Department, California State University Fresno, Fresno, CA 93740, USA and Project IDEI, University "Al. I. Cuza" Iaşi, Romania emails: sergiu.vacaru@gmail.com and dougs@csufrenso.edu June 28, 2017 Abstract Models of geometric flows pertaining to 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 Kählerian spaces M n = SU (1, 1+ k)/U (1) × 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 connec- tions 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. In particular, we analyze small parametric deformations when the underlying scale symmetry is preserved and the nontrivial anisotropic vacuum corresponds to generalized de Sitter spaces. Such configurations may mimic quantum effects whenever transitions to flat space are possible. Our approach allows us to generate solutions with scale violating terms induced by geometric flows, off–diagonal interactions and supersymmetric modifications of effective potentials. We study reconstructing the formalism for (super) geometric flows and modified gravity cosmological scenarios. We also analyze the conditions under which such modified mimetic type theories and solutions reproduce the Starobinsky inflationary models in the double scalar approach. Keywords: supergeometric flows, super Ricci solitons, modified (super) gravity, Perelman’s superfunctionals, exact solutions in (super) gravity, anisotropic cosmological solutions and inflation. ∗ Address for post correspondence: 67 Lloyds Street South, Manchester, M14 7LF, the UK; the USA affiliation is for a visiting position 1