Journal of the Mathematical Society of Japan

Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics

Francis E. BURSTALL, Udo HERTRICH-JEROMIN, and Yoshihiko SUYAMA

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Abstract

There is a one-to-one correspondence between associated families of generic conformally flat (local-)hypersurfaces in 4-dimensional space forms and conformally flat 3-metrics with the Guichard condition. In this paper, we study the space of conformally flat 3-metrics with the Guichard condition: for a conformally flat 3-metric with the Guichard condition in the interior of the space, an evolution of orthogonal (local-)Riemannian 2-metrics with constant Gauss curvature $-1$ is determined; for a 2-metric belonging to a certain class of orthogonal analytic 2-metrics with constant Gauss curvature $-1$, a one-parameter family of conformally flat 3-metrics with the Guichard condition is determined as evolutions issuing from the 2-metric.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 2 (2018), 617-649.

Dates
First available in Project Euclid: 18 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1524038668

Digital Object Identifier
doi:10.2969/jmsj/07027420

Mathematical Reviews number (MathSciNet)
MR3787734

Zentralblatt MATH identifier
06902436

Subjects
Primary: 53B25: Local submanifolds [See also 53C40]
Secondary: 53A30: Conformal differential geometry

Keywords
conformally flat hypersurface surface metric with constant Gauss curvature $-1$ Guichard net system of evolution equations

Citation

BURSTALL, Francis E.; HERTRICH-JEROMIN, Udo; SUYAMA, Yoshihiko. Curvilinear coordinates on generic conformally flat hypersurfaces and constant curvature 2-metrics. J. Math. Soc. Japan 70 (2018), no. 2, 617--649. doi:10.2969/jmsj/07027420. https://projecteuclid.org/euclid.jmsj/1524038668


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