An analog of Chern's conjecture for the Euler-Satake characteristic of affine orbifolds

JOURNAL OF GEOMETRY AND PHYSICS

卷:142

Article

Bagaev, Andrey V.¹  Zhukova, Nina I.² 

[1] Nizhnii Novgorod State Tech Univ, Minina St 24, Nizhnii Novgorod 603000, Russia

[2] Natl Res Univ Higher Sch Econ, Dept Informat Math & Comp Sci, Ul Bolshaja Pecherskaja 25-12, Nizhnii Novgorod 603155, Russia

关键词: Orbifold;    Euler characteristic of a manifold;    Euler-Satake characteristic of an orbifold;    Affine orbifold;    Special affine orbifold;   

DOI  :  10.1016/j.geomphys.2019.04.002

来源: Elsevier

【 摘 要 】

S.S. Chern conjectured that the Euler characteristic of every closed affine manifold has to vanish. We present an analog of this conjecture stating that the Euler-Satake characteristic of any compact affine orbifold is equal to zero. We prove that Chern's conjecture is equivalent to its analog for the Euler-Satake characteristic of compact affine orbifolds, not necessarily effective. This fact allowed us to extend to orbifolds sufficient conditions for Chern's conjecture proved by Klingler and Kostant-Sullivan. Thus, we prove that, if an n-dimensional compact affine orbifold N is complete or if its holonomy group belongs to the special linear group SL(n, R), then the Euler-Satake characteristic of N has to vanish. An application to pseudo-Riemannian orbifolds is considered. We give examples of orbifolds from the class under investigation. In particular, we construct an example of a compact incomplete affine orbifold with vanishing Euler-Satake characteristic, the holonomy group of which is not contained in SL(n, R). (C) 2019 Published by Elsevier B.V.

【 授权许可】

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