Projectively equivalent metrics subject to constraints

Author:
William Taber

Journal:
Trans. Amer. Math. Soc. **282** (1984), 711-737

MSC:
Primary 53C40; Secondary 53C50

DOI:
https://doi.org/10.1090/S0002-9947-1984-0732115-8

MathSciNet review:
732115

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Abstract: This work examines the relationship between pairs of projectively equivalent Riemannian or Lorentz metrics and on a manifold that induce the same Riemannian metric on a hypersurface . In general such metrics must be equal. In the case of distinct metrics, the structure of the metrics and the manifold are strongly determined by the set, , of points at which and are conformally related. The space is locally a warped product manifold over the hypersurface . In the Lorentz setting, is empty. In the Riemannian case, contains at most two points. If is nonempty, then is isometric to a standard sphere. Furthermore, in the case that contains one point, natural hypotheses imply is diffeomorphic to . If contains two points is diffeomorphic to .

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DOI:
https://doi.org/10.1090/S0002-9947-1984-0732115-8

Article copyright:
© Copyright 1984
American Mathematical Society