Note on the linear endomorphisms of a vector bundle

Abstract

In this paper, it is shown that if the Lie algebras of linear endomorphism fields of two vector bundles E and E′ are isomorphic, then E′ is isomorphic to E⊗L for some line bundle L. Analogous results are obtained for the endomorphism fields of vector bundles equipped with some structures (Riemann structure, symplectic structure, ...).