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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A new variational characterization of $n$-dimensional space forms


Authors: Zejun Hu and Haizhong Li
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3005-3023
MSC (2000): Primary 53C20, 53C25
Published electronically: December 9, 2003
MathSciNet review: 2052939
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Abstract: A Riemannian manifold $(M^n,g)$ is associated with a Schouten $(0,2)$-tensor $C_g$ which is a naturally defined Codazzi tensor in case $(M^n,g)$ is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional $\mathcal{F}_k[g]=\int_M\sigma_k(C_g)dvol_g$ defined on $\mathcal{M}_1=\{g\in\mathcal{M}\vert Vol(g)=1\}$, where $\mathcal{M}$ is the space of smooth Riemannian metrics on a compact smooth manifold $M$ and $\{\sigma_k(C_g), 1\leq k\leq n\}$ is the elementary symmetric functions of the eigenvalues of $C_g$ with respect to $g$. We prove that if $n\geq 5$ and a conformally flat metric $g$ is a critical point of $\mathcal{F}_2\vert _{\mathcal{M}_1}$ with $\mathcal{F}_2[g]\geq0$, then $g$ must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of $\mathcal{F}_2\vert _{\mathcal{M}_1}$ with $\mathcal{F}_2[g]\geq0$ characterized the three-dimensional space forms.


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Additional Information

Zejun Hu
Affiliation: Department of Mathematics, Zhengzhou University, Zhengzhou 450052, People’s Republic of China
Email: huzj@zzu.edu.cn

Haizhong Li
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
Email: hli@math.tsinghua.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03486-X
PII: S 0002-9947(03)03486-X
Keywords: Locally conformally flat Riemannian manifold, Schouten tensor, space form, Riemannian functional
Received by editor(s): September 30, 2002
Published electronically: December 9, 2003
Additional Notes: The first author was partially supported by grants from CSC, NSFC and the Outstanding Youth Foundation of Henan, China.
The second author was partially supported by the Alexander von Humboldt Stiftung and Zhongdian grant of NSFC
Article copyright: © Copyright 2003 American Mathematical Society