  We study inverse boundary problems for the magnetic Schr\"odinger operator
with H\"older continuous magnetic potentials and continuous electric potentials
on a conformally transversally anisotropic Riemannian manifold of dimension n
greater than or equal to three with connected boundary. A global uniqueness
result is established for magnetic fields and electric potentials from the
partial Cauchy data on the boundary of the manifold provided that the geodesic
X-ray transform on the transversal manifold is injective.
