Projectively equivalent metrics subject to constraints
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- by William Taber PDF
- Trans. Amer. Math. Soc. 282 (1984), 711-737 Request permission
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 ${S^n}$.References
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Additional Information
- © Copyright 1984 American Mathematical Society
- 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