Complex and symplectic geometry of vector bundle manifolds

The aim of this paper is to explore the complex and symplectic geometries of vector bundle manifolds. We will construct an almost complex structure on total spaces of vector bundles, endowed with a complex structure, over an almost complex base. Then we give necessary and sufficient conditions for its integrability. Meanwhile, we accomplish a symplectic version of this construction. We construct almost symplectic structures on vector bundle manifolds and we characterize those which are symplectic on the total space. Finally, we apply the constructions to the case of tangent bundles and Whitney sums. In particular, we obtain an infinite family of non-compact flat Kähler manifolds.