In Mirror Symmetry, specially in Homological Mirror Symmetry, the mirror of stable holomorphic Vector bundle is the special Lagrangian submanifold. In symplectic geometry A-brane = Lagrangian submanifold + flat vector bundle • In holomorphic geometry B-brane = complex submanifold + holomorphic bundle There is a relation between reflexive Sheaf and Vector bundle which is used in the study of Extension theory for finding canonical metrics A reflexive sheaf $ F$, on Kähler variety $X$ outside of codimension at least 3 is a holomorphic vector bundle, In the study of positivity theory of direct image of Line bundles which are Vector bundles. For example positivity of CM-bundle which is related to K-stability and Kähler-Einstein metric on Fano manifolds