On the RR Lyrae stars in globulars: IV. $\omega$ Centauri Optical UBVRI
  Photometry
  New accurate and homogeneous optical UBVRI photometry has been obtained for
variable stars in the Galactic globular $\omega$ Cen (NGC 5139). We secured
8202 CCD images covering a time interval of 24 years and a sky area of 84x48
arcmin. The current data were complemented with data available in the
literature and provided new, homogeneous pulsation parameters (mean magnitudes,
luminosity amplitudes,periods) for 187 candidate $\omega$ Cen RR Lyrae (RRLs).
Among them we have 101 RRc (first overtone), 85 RRab (fundamental) and a single
candidate RRd (double-mode) variables. Candidate Blazhko RRLs show periods and
colors that are intermediate between RRc and RRab variables, suggesting that
they are transitional objects.
  The comparison of the period distribution and of the Bailey diagram indicates
that RRLs in $\omega$ Cen show a long-period tail not present in typical
Oosterhoff II (OoII) globulars. The RRLs in dwarf spheroidals and in ultra
faint dwarfs have properties between Oosterhoff intermediate and OoII clusters.
Metallicity plays a key role in shaping the above evidence. These findings do
not support the hypothesis that $\omega$ Cen is the core remnant of a spoiled
dwarf galaxy.
  Using optical Period-Wesenheit relations that are reddening-free and
minimally dependent on metallicity we find a mean distance to $\omega$ Cen of
13.71$\pm$0.08$\pm$0.01 mag (semi-empirical and theoretical calibrations).
Finally, we invert the I-band Period-Luminosity-Metallicity relation to
estimate individual RRLs metal abundances. The metallicity distribution agrees
quite well with spectroscopic and photometric metallicity estimates available
in the literature.
