Positivity of Riemannian metric and the existence theorem of $L^2$ estimates for $d$ operator

Apply the $L^2$ technique developed by Deng-Ning-Wang-Zhou,we give a new characterization of Nakano $q$-positivity for Riemannian flat vector bundle in the sense of $L^2$ estimate for the $d$ operator. We also give a new characterization of the positivity of the curvature operator associated to the Riemannian metric on manifold in a local version in the sense of $L^2$ estimate. Then we prove two results parallel to Liu-Yang-Zhou’s answer to Lempert’s question to Hermitian holomorphic vector bundle. We also study the boundary convexity of relative compact domain in certain Riemannian manifold with convexity restrictions.