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We give a characterization of Nakano positivity of Riemannian flat vectorbundles over bounded domains $D\subset\mathbb{R}^n$ in terms of solvability ofthe $d$ equation with certain good $L^2$ estimate condition. As an application,we give an alternative proof of the matrix-valued Prekopa's theorem that isoriginally proved by Raufi. Our methods are inspired by the recent works ofDeng-Ning-Wang-Zhou on characterization of Nakano positivity of Hermitianholomorphic vector bundles and positivity of direct image sheaves associated toholomorphic fibrations.

Characterization of Curvature positivity of Riemannian metrics on flat vector bundles