On the RR Lyrae stars in globulars: IV. ω 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.