Under some positivity assumptions, extension properties of rationally
connected fibrations from a submanifold to its ambient variety are studied.
Given a family of rational curves on a complex projective manifold X inducing a
covering family on a submanifold Y with ample normal bundle in X, the main
results relate, under suitable conditions, the associated rational connected
fiber structures on X and on Y. Applications of these results include an
extension theorem for Mori contractions of fiber type and a classification
theorem in the case Y has a structure of projective bundle or quadric
fibration.