  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.
