Euler characteristic numbers of space-like manifolds

TL;DR

This paper proves a positivity result for the Euler characteristic of certain negatively curved manifolds homotopic to space-like manifolds and confirms Gromov's minimal volume conjecture for all compact even-dimensional space-like manifolds.

Contribution

It establishes a new topological constraint on negatively curved manifolds and verifies Gromov's conjecture in the context of space-like manifolds.

Findings

Euler characteristic satisfies a sign condition for negatively curved manifolds

Gromov's minimal volume conjecture holds for all compact even-dimensional space-like manifolds

Provides new insights into the topology 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.