  We show that the horizon geometry for supersymmetric black hole solutions of
minimal five-dimensional gauged supergravity is that of a particular
Einstein-Cartan-Weyl (ECW) structure in three dimensions, involving the trace
and traceless part of both torsion and nonmetricity, and obeying some precise
constraints. In the limit of zero cosmological constant, the set of nonlinear
partial differential equations characterizing this ECW structure reduces
correctly to that of a hyper-CR Einstein-Weyl structure in the Gauduchon gauge,
which was shown by Dunajski, Gutowski and Sabra to be the horizon geometry in
the ungauged BPS case.
