Asian Journal of Mathematics

Volume 21 (2017)

Number 3

Euler characteristic numbers of space-like manifolds

Pages: 591 – 598



Bing-Long Chen (Department of Mathematics, Sun Yat-Sen University, Guangzhou, China)

Kun Zhang (Department of Mathematics, South China University of Technology, Guangzhou, China)


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


Euler characteristic, Hopf conjecture

2010 Mathematics Subject Classification

35R01, 53C05

Received 10 October 2013

Published 5 July 2017