  The "VISTA near-infrared YJKs survey of the Magellanic System" (VMC) is
collecting deep Ks-band time-series photometry of pulsating stars hosted by the
two Magellanic Clouds and their connecting Bridge. Here we present YJKs light
curves for a sample of 717 Small Magellanic Cloud (SMC) Classical Cepheids
(CCs). These data, complemented with our previous results and V magnitude from
literature, allowed us to construct a variety of period-luminosity and
period-Wesenheit relationships, valid for Fundamental, First and Second
Overtone pulsators. These relations provide accurate individual distances to
CCs in the SMC over an area of more than 40 sq. deg. Adopting literature
relations, we estimated ages and metallicities for the majority of the
investigated pulsators, finding that: i) the age distribution is bimodal, with
two peaks at 120+-10 and 220+-10 Myr; ii) the more metal-rich CCs appear to be
located closer to the centre of the galaxy. Our results show that the
three-dimensional distribution of the CCs in the SMC, is not planar but heavily
elongated for more than 25-30 kpc approximately in the east/north-east towards
south-west direction. The young and old CCs in the SMC show a different
geometric distribution. Our data support the current theoretical scenario
predicting a close encounter or a direct collision between the Clouds some 200
Myr ago and confirm the presence of a Counter-Bridge predicted by some models.
The high precision three-dimensional distribution of young stars presented in
this paper provides a new testbed for future models exploring the formation and
evolution of the Magellanic System.
