  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
$(-1)^{\frac{n}{2}}\chi(M^{n})>0$. We also show that the minimal volume
conjecture of Gromov is true for all compact even dimensional space-like
manifolds.
