INVARIANT MEASURE AND THE EULER CHARACTERISTIC OF PROJECTIVELY ELAT MANIFOLDS

Title & Authors
INVARIANT MEASURE AND THE EULER CHARACTERISTIC OF PROJECTIVELY ELAT MANIFOLDS
Jo, Kyeong-Hee; Kim, Hyuk;

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 RP$\small{^{n}}$ invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chen's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on RP$\small{^{n}}$ ; 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;
Language
English
Cited by
1.
The analytic continuation of hyperbolic space, Geometriae Dedicata, 2012, 161, 1, 129
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