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