  We consider integral geometry inverse problems for unitary connections and
skew-Hermitian Higgs fields on manifolds with negative sectional curvature. The
results apply to manifolds in any dimension, with or without boundary, and also
in the presence of trapped geodesics. In the boundary case, we show injectivity
of the attenuated ray transform on tensor fields with values in a Hermitian
bundle (i.e. vector valued case). We also show that a connection and Higgs
field on a Hermitian bundle are determined up to gauge by the knowledge of the
parallel transport between boundary points along all possible geodesics. The
main tools are an energy identity, the Pestov identity with a unitary
connection, which is presented in a general form, and a precise analysis of the
singularities of solutions of transport equations when there are trapped
geodesics. In the case of closed manifolds, we obtain similar results modulo
the obstruction given by twisted conformal Killing tensors, and we also study
this obstruction.
