  We study the impact of the assumption of a non-flat fiducial cosmology on the
measurement, analysis and interpretation of BAO distance variables, along and
across the line-of-sight. The assumption about cosmology enters in the choice
of the base template, as well as on the transformation of tracer's redshifts
into distances (the catalog cosmology): here we focus on the curvature
assumption, separately and jointly, on both. We employ BOSS and eBOSS publicly
available data and show that for the statistical precision of this data set,
distance measures and thus cosmological inference are robust to assumptions
about curvature both of the template and the catalog. Thus the usual
assumptions of flat fiducial cosmologies (but also assumptions of non-flat
cosmologies) do not produce any detectable systematic effects. For forthcoming
large-volume surveys, however, small but appreciable residual systematic shifts
can be generated which may require some care. These are mostly driven by the
choice of catalog cosmology if it is significantly different from true
cosmology. In particular, the catalog (and template) cosmology should be
chosen, possibly iteratively, in such a way that the recovered BAO scaling
variables are sufficiently close to unity. At this level of precision, however,
other previously overlooked effects become relevant, such as a mismatch between
the sound horizon as seen in the BAO and the actual sound horizon in the early
Universe. If unaccounted for, such effect may be misinterpreted as cosmological
and thus bias the curvature (and cosmology) constraints. We present best
practices to avoid this.
