  In this work we present the foundations of generalized scalar-tensor theories
arising from vector bundle constructions, and we study the kinematic, dynamical
and cosmological consequences. In particular, over a pseudo-Riemannian
space-time base manifold, we define a fiber structure with two scalar fields.
The resulting space is a 6-dimensional vector bundle endowed with a non-linear
connection. We provide the form of the geodesics and the Raychaudhuri and
general field equations, both in Palatini and metrical method. When applied at
a cosmological framework, this novel geometrical structure induces extra terms
in the modified Friedmann equations, leading to the appearance of an effective
dark energy sector, as well as of an interaction of the dark mater sector with
the metric. We show that we can obtain the standard thermal history of the
universe, with the sequence of matter and dark-energy epochs, and furthermore
the effective dark-energy equation-of-state parameter can lie in the
quintessence or phantom regimes, or exhibit the phantom-divide crossing.
