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21 votes

For which $b$ it is possible that $S^n$ can have a Lorentz metric? Why?

I would just like to point out that a slightly more general version of the statement in Moroianu's answer holds: A compact orientable manifold $M$ admits a Lorentzian metric if and only if $\chi(M)=0 …
Renato G. Bettiol's user avatar