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# Asian Journal of Mathematics

## Volume 21 (2017)

### Number 3

### Euler characteristic numbers of space-like manifolds

Pages: 591 – 598

DOI: http://dx.doi.org/10.4310/AJM.2017.v21.n3.a9

#### Authors

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

#### Keywords

Euler characteristic, Hopf conjecture

#### 2010 Mathematics Subject Classification

35R01, 53C05

Received 10 October 2013

Published 5 July 2017