Invariant Measure and the Euler Characteristic of Projectively Flat Manifolds

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 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 RP^n; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.