[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2003.40.1.109

INVARIANT MEASURE AND THE EULER CHARACTERISTIC OF PROJECTIVELY ELAT MANIFOLDS

Jo, Kyeong-Hee (School of Mathematical Sciences Seoul National University)
Kim, Hyuk (School of Mathematical Sciences Seoul National University)

Publication Information

Journal of the Korean Mathematical Society / v.40, no.1, 2003 , pp. 109-128 More about this Journal

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 $^{n}$ 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 $^{n}$ ; 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;

Citations & Related Records

Times Cited By Web Of Science : 0 (Related Records In Web of Science)
Times Cited By SCOPUS : 0

Reference

1	The Euler characteristic of a certain class of projectively flat manifolds / [ H. Kim;H. Lee ] / Topology Appl. DOI ScienceOn
2	/ [ N. Dunford;J. T. Schwartz ] / Linear Operators
3	Foliated bundles, invariant measures and flat manifolds / [ M. W. Hirsch;W. P. Thurston ] / Ann. of Math.
4	Geometric structures on manifolds and varieties of representations / [ W. M. Goldman ] / Contemp. Math. DOI
5	The affine structures on the real two-torus / [ T. Nagano;K. Yagi ] / Osaka J. Math.
6	Invariant distances for projective structures / [ S. Kobayashi ] / Ist. Naz. Alta. Mat. Symp. Math.
7	Invariant means on topological groups and their applications / [ F. P. Greenleaf ] / Van Nostrand Math. Studies #16
8	Closed similarity manifolds / [ D. Fried ] / Comment. Math. Helv. DOI
9	Affine manifolds with nilpotent holonomy / [ D. Fried;W. Goldman;M. Hirsch ] / Comm. Math. Helv. DOI
10	On fundamental groups of complete affinely flat manifolds / [ J. Milnor ] / Adv. Math. DOI
11	Ergodic theory and semisimple groups / [ R. J. Zimmer ] / Monographs in Mathematics.
12	A Polyhedral Gauss-Bonnet formula and projectively flat manifolds / [ H. Kim ] / Proceedings of the international conference on pure and applied Math., Beijing and Yanji
13	/ [ W. P. Thurston ] / The geometry and topology of 3-manifolds
14	The Euler characteristic of a compact affine space form is zero / [ B. Kostant;D. Sullivan ] / Bull. Amer. Math. Soc. DOI
15	Differential Geometry in the Large / [ H. Hopf ] / Lecture notes in Math.
16	The Euler characteristic of projectively flat manifolds with amenable fundamental groups / [ H. Kim;H. Lee ] / Proc. Amer. Math. Soc. DOI ScienceOn