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In this work, we study the finite time blow-up phenomenon of three types ofsemilinear wave systems with multiple speeds, posed on asymptotically Euclideanmanifolds. We establish the upper bound estimates for the lifespan of solutionswhen the spatial dimension $n\geq 2$. In particular, for system related to theGlassey conjecture, we obtained the finite time blow up results under a newcurve. This new curve is sharp at least for $n=3$.

Finite time blow up for systems of wave equations with multiple speeds posed on asymptotically Euclidean manifold