  Recently Berman and Perry constructed a four-dimensional M-theory effective
action which manifests SL(5) U-duality. Here we propose an underlying
differential geometry of it, under the name `SL(5) U-geometry' which
generalizes the ordinary Riemannian geometry in an SL(5) compatible manner. We
introduce a `semi-covariant' derivative that can be converted into fully
covariant derivatives after anti-symmetrizing or contracting the SL(5) vector
indices appropriately. We also derive fully covariant scalar and Ricci-like
curvatures which constitute the effective action as well as the equation of
motion.
