The hopf conjecture for manifolds with abelian group actions

Xiaochun Rong; S. U. Xiaole

doi:10.1142/S0219199705001660

The hopf conjecture for manifolds with abelian group actions

Xiaochun Rong, S. U. Xiaole

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

Let M be a closed even n-manifold of positive sectional curvature on which a torus T^k acts isometrically. We show that if k ≥ n-4/8 (respectively, k > 1) for n ≠ 12 (respectively, n = 12), then the Euler characteristic of each T^k-fixed point component is positive. This implies that the Euler characteristic of M is positive. We also extend this result to an isometric elementary p-group ℤ_p^k-action on a closed manifold of positive sectional curvature.

Original language	English (US)
Pages (from-to)	121-136
Number of pages	16
Journal	Communications in Contemporary Mathematics
Volume	7
Issue number	1
DOIs	https://doi.org/10.1142/S0219199705001660
State	Published - Feb 2005

All Science Journal Classification (ASJC) codes

Mathematics(all)
Applied Mathematics

Keywords

Euler characteristics
Hopf conjecture
Positive curvature
Torus action

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10.1142/S0219199705001660

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title = "The hopf conjecture for manifolds with abelian group actions",

abstract = "Let M be a closed even n-manifold of positive sectional curvature on which a torus Tk acts isometrically. We show that if k ≥ n-4/8 (respectively, k > 1) for n ≠ 12 (respectively, n = 12), then the Euler characteristic of each Tk-fixed point component is positive. This implies that the Euler characteristic of M is positive. We also extend this result to an isometric elementary p-group ℤpk-action on a closed manifold of positive sectional curvature.",

keywords = "Euler characteristics, Hopf conjecture, Positive curvature, Torus action",

author = "Xiaochun Rong and Xiaole, {S. U.}",

note = "Funding Information: The First author was supported partially by NSF Grant DMS 0203164 and by a reach grant from Capital normal university. Funding Information: The Second author was supported partially by grant NSFC (in China) 10371008.",

year = "2005",

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doi = "10.1142/S0219199705001660",

language = "English (US)",

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T1 - The hopf conjecture for manifolds with abelian group actions

AU - Rong, Xiaochun

AU - Xiaole, S. U.

N1 - Funding Information: The First author was supported partially by NSF Grant DMS 0203164 and by a reach grant from Capital normal university. Funding Information: The Second author was supported partially by grant NSFC (in China) 10371008.

PY - 2005/2

Y1 - 2005/2

N2 - Let M be a closed even n-manifold of positive sectional curvature on which a torus Tk acts isometrically. We show that if k ≥ n-4/8 (respectively, k > 1) for n ≠ 12 (respectively, n = 12), then the Euler characteristic of each Tk-fixed point component is positive. This implies that the Euler characteristic of M is positive. We also extend this result to an isometric elementary p-group ℤpk-action on a closed manifold of positive sectional curvature.

AB - Let M be a closed even n-manifold of positive sectional curvature on which a torus Tk acts isometrically. We show that if k ≥ n-4/8 (respectively, k > 1) for n ≠ 12 (respectively, n = 12), then the Euler characteristic of each Tk-fixed point component is positive. This implies that the Euler characteristic of M is positive. We also extend this result to an isometric elementary p-group ℤpk-action on a closed manifold of positive sectional curvature.

KW - Euler characteristics

KW - Hopf conjecture

KW - Positive curvature

KW - Torus action

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EP - 136

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

IS - 1

ER -

The hopf conjecture for manifolds with abelian group actions

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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