= Parallel metrics with a common Levi-Civita connection
{title2=$\widetilde g(u,v)=g(Au,v),\quad\nabla A=0$}
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.
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