Nope! :) That's the point of the "locally conserved" clarification I made in a sibling comment: these conservation laws hold at every individual point in space,

Nope! :) That's the point of the "locally conserved" clarification I made in a sibling comment: these conservation laws hold at every individual point in space, regardless of whether the point is part of a curved manifold or a flat one. (Curvature can definitely interfere with attempts to define a global conservation law, like "total momentum in the universe is constant". But the essential feature of a manifold is that near any given point it is locally equivalent to simple flat space.)