  In this paper we consider the lens rigidity problem with partial data for
conformal metrics in the presence of a magnetic field on a compact manifold of
dimension $\geq 3$ with boundary. We show that one can uniquely determine the
conformal factor and the magnetic field near a strictly convex (with respect to
the magnetic geodesics) boundary point where the lens data is accessible. We
also prove a boundary rigidity result with partial data assuming the lengths of
magnetic geodesics joining boundary points near a strictly convex boundary
point are known. The local lens rigidity result also leads to a global rigidity
result under some strictly convex foliation condition. A discussion of a weaker
version of the lens rigidity problem with partial data for general smooth
curves is given at the end of the paper.
