  This paper develops new techniques for studying smooth dynamical systems in
the presence of a \CC metric. Principally, we employ the theory of
Margulis-Mostow, M\'etivier, Mitchell and Pansu on tangent cones to establish
resonances between Lyapunov exponents. We apply these results in three
different settings. First, we explore rigidity properties of smooth dominated
splittings for Anosov diffeomorphisms and flows via associated smooth \CC
metrics. Second, we obtain local rigidity properties of higher hyperbolic rank
metrics in a neighborhood of a locally symmetric one. For the latter
application we also prove structural stability of the Brin-Pesin asymptotic
holonomy group for frame flows. Finally, we obtain local rigidity properties
for uniform lattice actions on the ideal boundary of quaternionic and
octonionic symmetric spaces.
