  The initial and boundary conditions of the Galactic star formation in
molecular clouds are not well understood. In an effort to shed new light on
this long-standing problem, we measured properties of dense cores and
filamentary structures in the Vela C molecular cloud, observed with Herschel.
We applied the getsf extraction method to separate the components of sources
and filaments from each other and their backgrounds, before detecting,
measuring, and cataloging the structures. The cores and filamentary structures
constitute 40% of the total mass of Vela C, most of the material is in the
low-density molecular background cloud. We selected 570 reliable cores, of
which 149 are the protostellar cores and 421 are the starless cores. Almost 78%
of the starless cores were identified with the gravitationally bound prestellar
cores. The exponent of the CMF (alpha = 1.35) is identical to that of the
Salpeter IMF. We selected 68 filaments with at least one side that appeared not
blended with adjacent structures. The filament widths are in the range of 0.15
pc to 0.63 pc, and have a median value of W = 0.3(0.11) pc. The surface
densities of filaments are well correlated with their contrasts and linear
densities. Within uncertainties of the filament instability criterion, many
filaments may well be both supercritical and subcritical. A large fraction of
filaments may definitely be considered supercritical, in which are found 94
prestellar cores, 83 protostellar cores, and only 1 unbound starless core.
Taking into account the uncertainties, the supercritical filaments contain only
prestellar and protostellar cores. Our findings support the idea that there
exists a direct relationship between the CMF and IMF and that filaments play a
key role in the formation of prestellar cores, which is consistent with the
previous Herschel results.
