  In the relativistic impulse approximation (RIA), we study symmetry potentials
and in-medium nucleon-nucleon (NN) cross sections with the Nambu-Jona-Lasinio
(NJL) model that features chiral symmetry. The chiral symmetry that plays a
fundamental role in the nonperturbative physics in the strong interaction is
anticipated to add restrictive effects on the symmetry potentials and in-medium
NN cross sections. For comparison, we also perform the study with the usual
relativistic mean-field (RMF) model. The numerical results with the NJL and RMF
models are similar at saturation density and below, since a priori fit was made
to saturation properties. With the increase of nuclear density, the chiral
symmetry starts to be restored partially in the NJL model, resulting in the
explicit fall of the scalar density. In a large energy span, the symmetry
potential acquires a significant rise for the partial restoration of the chiral
symmetry, compared to the one with the RMF model. It is found that the
in-medium NN cross sections in the RIA with the NJL and RMF models both
increase with the density in the energy region interested in this study,
whereas those with the NJL model increase sharply as long as a clear chiral
symmetry restoration takes place. The different tendency of observables in
density can be transmitted to the different energy dependence in the RIA. The
NJL model is shown to have characteristic energy-dependent symmetry potentials
and NN cross sections beyond saturation point, apart from the RMF models.
