  In the present work, we study linear, torsion-free metric-Palatini gravity,
extended by the quadratics of the antisymmetric part of the Ricci tensor and
extended also by the presence of the affine connection in the matter sector. We
show that this extended metric-Palatini gravity reduces dynamically to the
general relativity plus a geometrical massive vector field corresponding to
non-metricity of the connection. We also show that this geometric Proca field
couples to fermions universally. We derive static, spherically symmetric field
equations of this Einstein-geometric Proca theory. We study possibility of
black hole solutions by taking into account the presence of a dust distribution
that couples to the geometric Proca. Our analytical and numerical analyses show
that the presence of this dust worsens the possibility of horizon formation. We
briefly discuss possible roles of this universally-coupled geometric Proca in
the astrophysical and collider processes.
