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Spacetime with general linear vector distortion is introduced. Thus, thetorsion and the nonmetricity of the affine connection are assumed to beproportional to a vector field (and not its derivatives). The resultingtwo-parameter family of non-Riemannian geometries generalises the conformalWeyl geometry and some other interesting special cases. Taking into account theleading order quadratic curvature correction to the Einstein-Hilbert actionresults uniquely in the one-parameter extension of the Starobinsky inflationknown as the alpha-attractor. The most general quadratic curvature actionintroduces, in addition to the canonical vector kinetic term, novel ghost-freevector-tensor interactions.

Spacetimes with vector distortion: Inflation from generalised Weyl geometry