  We briefly review some basic concepts of parallel displacement in Finsler
geometry. In general relativity, the parallel translation of objects along the
congruence of the fundamental observer corresponds to the evolution in time. By
dropping the quadratic restriction on the measurement of an infinitesimal
distance, the geometry is generalized to a Finsler structure. Apart from
curvature a new property of the manifold complicates the geometrodynamics, the
color. The color brings forth an intrinsic local anisotropy and many quantities
depend on position and to a "supporting" direction. We discuss this direction
dependence and some physical interpretations. Also, we highlight that in
Finsler geometry the parallel displacement isn't necessarily always along the
"supporting" direction. The latter is a fundamental congruence of the manifold
and induces a natural 1+3 decomposition. Its internal deformation is given
through the evolution of the irreducible components of vorticity, shear and
expansion.
