  Consider a compact Riemannian manifold of dimension $\geq 3$ with strictly
convex boundary, such that the manifold admits a strictly convex function. We
show that the attenuated ray transform in the presence of an arbitrary
connection and Higgs field is injective modulo the natural obstruction for
functions and one-forms. We also show that the connection and the Higgs field
are uniquely determined by the scattering relation modulo gauge
transformations. The proofs involve a reduction to a local result showing that
the geodesic X-ray transform with a matrix weight can be inverted locally near
a point of strict convexity at the boundary, and a detailed analysis of layer
stripping arguments based on strictly convex exhaustion functions. As a
somewhat striking corollary, we show that these integral geometry problems can
be solved on strictly convex manifolds of dimension $\geq 3$ having
non-negative sectional curvature (similar results were known earlier in
negative sectional curvature). We also apply our methods to solve some inverse
problems in quantum state tomography and polarization tomography.
