  We derive, from the dimensional cross-over criterion, a fundamental-measure
density functional for parallel hard curved rectangles moving on a cylindrical
surface. We derive it from the density functional of circular arcs of length
$\sigma$ with centers of mass located on an external circumference of radius
$R_0$. The latter functional in turns is obtained from the corresponding 2D
functional for a fluid of hard discs of radius $R$ on a flat surface with
centers of mass confined onto a circumference of radius $R_0$. Thus the curved
length of closest approach between two centers of mass of hard discs on this
circumference is $\sigma=2R_0\sin^{-1}(R/R_0)$, the length of the circular
arcs. From the density functional of circular arcs, and by applying a
dimensional expansion procedure to the spatial dimension orthogonal to the
plane of the circumference, we finally obtain the density functional of curved
rectangles of edge-lengths $\sigma$ and $L$. The DF for curved rectangles can
also be obtained by fixing the centers of mass of parallel hard cylinders of
radius $R$ and length $L$ on a cylindrical surface of radius $R_0$. The phase
behavior of a fluid of aligned curved rectangles is obtained by calculating the
free-energy branches of smectic, columnar and crystalline phases for different
values of the ratio $R_0/R$ in the range $1<R_0/R\leq 4$; the smectic phase
turns out to be the most stable except for $R_0/R=4$ where the crystalline
phase becomes reentrant in a small range of packing fractions. When $R_0/R<1$
the transition is absent, since the density functional of curved rectangles
reduces to the 1D Percus functional.
