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Apply the $L^2$ technique developed by Deng-Ning-Wang-Zhou,we give a newcharacterization of Nakano $q$-positivity for Riemannian flat vector bundle inthe sense of $L^2 $ estimate for the $d$ operator. We also give a newcharacterization of the positivity of the curvature operator associated to theRiemannian 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 toLempert's question to Hermitian holomorphic vector bundle. We also study theboundary convexity of relative compact domain in certain Riemannian manifoldwith convexity restrictions.

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