Euler characteristic numbers of space-like manifolds

In this note, we prove that if a compact even dimensional manifold $M^{n}$ with negative sectional curvature is homotopic to some compact space-like manifold $N^{n}$, then the Euler characteristic number of $M^{n}$ satisfies $(-1)^{\frac{n}{2}}χ(M^{n})>0$. We also show that the minimal volume conjecture of Gromov is true for all compact even dimensional space-like manifolds.

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