Extension of holomorphic differential forms (Q2762953)

The author extends the well-known Hahn-Banach theorem for continuation of linear functionals on normed spaces to vector bundles on smooth manifolds. The notions of almost complex, almost quaternion and almost octonion manifolds are introduced consecutively. Then a theorem on extension of differential 1-forms on a manifold holomorphic with respect to one, two or three almost complex structures is proved under the requirement that the manifold is a submanifold of a Stein manifold with respect to one of these structures.