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A new variational characterization of three-dimensional space forms. (English) Zbl 1006.58008
The authors consider a generalization of the concept of Riemannian metric on a manifold, the so-called Weyl structure, that is a class of conformal Riemannian metrics on the manifold and a torsion-free connection preserving the conformal structure. It is proved that the set of Weyl structures can be equipped with a structure of a Banach manifold and some functionals (involving the curvature of the Weyl structures) on such an infinite dimensional manifold are considered. Critical points of these functionals are studied and in some case, the critical points are characterized as Einstein-Weyl structures, exactly as in Riemannian Geometry, Einstein metrics are the critical points of suitable functionals involving the scalar curvature.

MSC:
58E11 Critical metrics
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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