Invariant Measure and the Euler Characteristic of Projectively Flat Manifolds

2002/10/22 by Kyeonghee Jo, Jo, Kyeonghee, Hyuk Kim +1
Mathematics · #53C15 #57R20 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #math.GT #msc:53C15 #msc:57R20

paper · pdf · doi:10.48550/arxiv.math/0210335

arxiv created 2002/10/22 · openalex publication_date 2002/10/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 RPⁿ 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 RPⁿ; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

Related

• Proof of Chern conjecture for flat affine manifolds
• Lectures on the Euler characteristic of affine manifolds
• Proof of Chern's conjecture on affine manifolds
• The generalized Chern conjecture for manifolds that are locally a product of surfaces
• Minimal Euler Characteristics for Even-Dimensional Manifolds with Finite Fundamental Group