  The initial shear field, characterized by a primordial perturbation
potential, plays a crucial role in the formation of large scale structures.
Hence, considerable analytic work has been based on the joint distribution of
its eigenvalues, associated with Gaussian statistics. In addition, directly
related morphological quantities such as ellipticity or prolateness are
essential tools in understanding the formation and structural properties of
halos, voids, sheets and filaments, their relation with the local environment,
and the geometrical and dynamical classification of the cosmic web. To date,
most analytic work has been focused on Doroshkevich's unconditional formulae
for the eigenvalues of the linear tidal field, which neglect the fact that
halos (voids) may correspond to maxima (minima) of the density field. I present
here new formulae for the constrained eigenvalues of the initial shear field
associated with Gaussian statistics, which include the fact that those
eigenvalues are related to regions where the source of the displacement is
positive (negative): this is achieved by requiring the Hessian matrix of the
displacement field to be positive (negative) definite. The new conditional
formulae naturally reduce to Doroshkevich's unconditional relations, in the
limit of no correlation between the potential and the density fields. As a
direct application, I derive the individual conditional distributions of
eigenvalues and point out the connection with previous literature. Finally, I
outline other possible theoretically- or observationally-oriented uses, ranging
from studies of halo and void triaxial formation, development of
structure-finding algorithms for the morphology and topology of the cosmic web,
till an accurate mapping of the gravitational potential environment of galaxies
from current and future generation galaxy redshift surveys.
