  Spacetime with general linear vector distortion is introduced. Thus, the
torsion and the nonmetricity of the affine connection are assumed to be
proportional to a vector field (and not its derivatives). The resulting
two-parameter family of non-Riemannian geometries generalises the conformal
Weyl geometry and some other interesting special cases. Taking into account the
leading order quadratic curvature correction to the Einstein-Hilbert action
results uniquely in the one-parameter extension of the Starobinsky inflation
known as the alpha-attractor. The most general quadratic curvature action
introduces, in addition to the canonical vector kinetic term, novel ghost-free
vector-tensor interactions.
