  The level curvature distribution function is studied both analytically and
numerically for the case of T-breaking perturbations over the orthogonal
ensemble. The leading correction to the shape of the curvature distribution
beyond the random matrix theory is calculated using the nonlinear
supersymmetric sigma-model and compared to numerical simulations on the
Anderson model. It is predicted analytically and confirmed numerically that the
sign of the correction is different for T-breaking perturbations caused by a
constant vector-potential equivalent to a phase twist in the boundary
conditions, and those caused by a random magnetic field. In the former case it
is shown using a nonperturbative approach that quasi-localized states in weakly
disordered systems can cause the curvature distribution to be nonanalytic. In
$2d$ systems the distribution function $P(K)$ has a branching point at K=0 that
is related to the multifractality of the wave functions and thus should be a
generic feature of all critical eigenstates. A relationship between the
branching power and the multifractality exponent $d_{2}$ is suggested. Evidence
of the branch-cut singularity is found in numerical simulations in $2d$ systems
and at the Anderson transition point in $3d$ systems.
