Rationally cubic connected manifolds I: manifolds covered by lines

In this paper we study smooth complex projective polarized varieties (X,H) of dimension n ge 2 2 which admit a covering family V of rational curves of degree 3 with respect to H such that two general points of X may be joined by a curve parametrized by V , and such that there is a covering family of rational curves of H-degree one. We prove that the Picard number of these manifolds is at most three, and that, if equality holds, (X,H) has an adjunction theoretic scroll structure over a smooth variety.