  A consistent theory of quantum gravity (QG) at Planck scale almost sure
contains manifestations of Lorentz local symmetry violations (LV) which may be
detected at observable scales. This can be effectively described and classified
by models with nonlinear dispersions and related Finsler metrics and
fundamental geometric objects (nonlinear and linear connections) depending on
velocity/ momentum variables. We prove that the trapping brane mechanism
provides an accurate description of gravitational and matter field phenomena
with LV over a wide range of distance scales and recovering in a systematic way
the general relativity (GR) and local Lorentz symmetries. In contrast to the
models with extra spacetime dimensions, the Einstein-Finsler type gravity
theories are positively with nontrivial nonlinear connection structure,
nonholonomic constraints and torsion induced by generic off-diagonal
coefficients of metrics, and determined by fundamental QG and/or LV effects.
