Journal of the
Korean Mathematical Society JKMS

In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on $\mathbb RP^n$ invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chern's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on $\mathbb RP^n$; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

Keywords: Euler characteristic, invariant measure, projectively flat manifold, affinely flat manifold, polyhedral Gauss-Bonnet formula, Chern's conjecture

MSC numbers: 57R20, 53C15