A new variational characterization of -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 | References | Similar Articles | Additional Information

Abstract: A Riemannian manifold is associated with a *Schouten* -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with 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:
https://doi.org/10.1090/S0002-9947-03-03486-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