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ISSN 1088-6826(e) ISSN 0002-9939(p)

A critical metric for the $L^2$ -norm of the curvature tensor on $S^3$

Author(s): François Lamontagne
Journal: Proc. Amer. Math. Soc. 126 (1998), 589-593.
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Abstract | References | Additional information

Abstract: The $L^2$ -norm of the curvature tensor

$\begin{displaymath}{\cal R}(g)=\frac{1}{(\mathrm{Vol}(M))^{\frac{n-4}{n}}}\int _{M}\mid R \mid ^{2}dvol_{g} \end{displaymath}$

defines a Riemannian functional on the space of metrics. This work exhibits a metric on $S^3$ which is of Berger type but not of constant ricci curvature, and yet is critical for ${\cal R}$ .

References:

1.: S. Helgason Differential Geometry, Lie Groups, and Symmetric Spaces Academic Press. MR 26:2986
2.: Francois Lamontagne Ph.D. Thesis S.U.N.Y. at Stony Brook.
3.: J. Milnor Curvature of Left Invariant Metrics on Lie Groups Advances in Mathematics 21 p.293-329 (1976). MR 54:12970
4.: R. Palais The Principle of Symmetric Criticality Communications in Mathematical Physics 69 p.19-30 (1979). MR 81c:58026

Additional Information:

François Lamontagne
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, Montréal, Québec, Canada
Email: lamontaf@crm.umontreal.ca

DOI: 10.1090/S0002-9939-98-04171-9
PII: S 0002-9939(98)04171-9
Keywords: Critical metrics, Hopf fibration, Berger sphere
Received by editor(s): December 7, 1995
Received by editor(s) in revised form: July 31, 1996
Communicated by: Christopher Croke
Copyright of article: Copyright 1998, American Mathematical Society

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