Euler characteristic numbers of space-like manifolds

Abstract

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.