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Let $p:\sXS$ be a proper K\"ahler fibration and $\sE\sX$ a Hermitianholomorphic vector bundle.  As motivated by the work of Berndtsson(\cite{Berndtsson09a}), by using basicHodge theory, we derive several general curvature formulas for the direct image$p_*(K_{\sX/S}\ts \sE)$ for general Hermitian holomorphic vector bundle $\sE$in a simple way.  A straightforward application is that, if the family $\sXS$ isinfinitesimally trivial and Hermitian vector bundle $\sE$ is Nakano-negativealong the base $S$, then the direct image $p_*(K_{\sX/S}\ts \sE)$ isNakano-negative. We also use these curvature formulas to study the moduli spaceof projectively flat vector bundles with positive first Chern classes andobtain that, if the Chern curvature of direct image $p_*(K_{X}\ts E)$--of apositive projectively flat family $(E,h(t))_{t\in \mathbb D}X$--vanishes, thenthe curvature forms of this family are connected by holomorphic automorphismsof the pair $(X,E)$.

Curvatures of direct image sheaves of vector bundles and applications I