Parallel metrics with a common Levi-Civita connection ($\tilde{g}(u,v)=g(Au,v),\nabla A=0$)

Two Riemannian metrics have the same Levi-Civita connection precisely when their relative positive self-adjoint endomorphism $A$ is parallel for that connection. Parallel transport preserves both metrics. Consequently agreement at one point implies agreement everywhere on a connected manifold. Without that agreement, constant scalar multiples always give examples; constant nonproportional Euclidean forms show that a common connection does not force scalar proportionality.