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Transactions of the American Mathematical Society

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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 $ g$ and $ {g^ \ast }$ on a manifold $ M$ that induce the same Riemannian metric on a hypersurface $ H$. 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, $ C$, of points at which $ g$ and $ {g^ \ast }$ are conformally related. The space $ (M - C,g)$ is locally a warped product manifold over the hypersurface $ H$. In the Lorentz setting, $ C$ is empty. In the Riemannian case, $ C$ contains at most two points. If $ C$ is nonempty, then $ H$ is isometric to a standard sphere. Furthermore, in the case that $ C$ contains one point, natural hypotheses imply $ M$ is diffeomorphic to $ {R^n}$. If $ C$ contains two points $ M$ is diffeomorphic to $ {S^n}$.


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DOI: https://doi.org/10.1090/S0002-9947-1984-0732115-8
Article copyright: © Copyright 1984 American Mathematical Society

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