  The local theory of the critical lines of 2D and 3D Gaussian fields that
underline the cosmic structures is presented. In the context of cosmological
matter distribution the subset of critical lines of the 3D density field serves
to delineate the skeleton of the observed filamentary structure at large
scales. A stiff approximation used to quantitatively describe the filamentary
skeleton shows that the flux of the skeleton lines is related to the average
Gaussian curvature of the 1D (2D) sections of the field, much in the same way
as the density of the peaks. The distribution of the length of the critical
lines with threshold is analyzed in detail, while the extended descriptors of
the skeleton - its curvature and its singular points, are introduced and
briefly described. Theoretical predictions are compared to measurements of the
skeleton in realizations of Gaussian random fields in 2D and 3D. It is found
that the stiff approximation predicts accurately the shape of the differential
length, allows for analytical insight, and explicit closed form solutions.
Finally, it provides a simple classification of the singular points of the
critical lines: i) critical points; ii) bifurcation points; iii) slopping
plateaux.
