Schrödinger connection with selfdual nonmetricity vector in 2+1 dimensions

We present a three-dimensional metric affine theory of gravity whose field equations lead to connection introduced by Schr\"odinger many decades ago. Although involving nonmetricity, the preserves length vectors under parallel transport, and appears thus be more physical than one proposed Weyl. By considering solutions with constant scalar curvature, we obtain self-duality relation for nonmetricity vector which implies Proca equation that may also interpreted in terms inhomogeneous Maxwell emerging from geometry.