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.