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