  The VISTA Variables in the Via Lactea (VVV) ESO Public Survey is opening a
new window to study the inner Galactic globular clusters using their variable
stars. These globular clusters have been neglected in the past due to the
difficulties caused by the presence of an elevated extinction and high field
stellar densities in their lines of sight. However, the discovery and study of
any present variables in these clusters, especially RRLyrae stars, can help to
greatly improve the accuracy of their physical parameters. It can also help to
shed some light on the interrogations brought by the intriguing Oosterhoff
dichotomy in the Galactic globular cluster system. In a series of papers we
plan to explore the variable stars in the globular clusters falling inside the
field of the VVV survey. In this first paper we search and study the variables
present in two highly-reddened, moderately metal-poor, faint, inner Galactic
globular clusters: 2MASS-GC02 and Terzan10. We report the discovery of sizable
populations of RR Lyrae stars in both globular clusters. We use near-infrared
period-luminosity relations to determine the color excess of each RR Lyrae
star, from which we obtain both accurate distances to the globular clusters and
the ratios of the selective to total extinction in their directions. We find
the extinction towards both clusters to be elevated, non-standard, and highly
differential. We also find both clusters to be closer to the Galactic center
than previously thought, with Terzan10 being on the far side of the Galactic
bulge. Finally, we discuss their Oosterhoff properties, and conclude that both
clusters stand out from the dichotomy followed by most Galactic globular
clusters.
