Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 58, Number 3 (2006), 903-925.
A Möbius characterization of submanifolds
In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere . First of all, we proved that, for an -dimensional submanifold without umbilical points and with vanishing Möbius form , if is satisfied, then, is Möbius equivalent to an open part of either the Riemannian product in , or the image of the conformal diffeomorphism of the standard cylinder in , or the image of the conformal diffeomorphism of the Riemannian product in , or is locally Möbius equivalent to the Veronese surface in . When , our pinching condition is the same as in Main Theorem of Hu and Li , in which they assumed that is compact and the Möbius scalar curvature is constant. Secondly, we consider the Möbius sectional curvature of the immersion . We obtained that, for an -dimensional compact submanifold without umbilical points and with vanishing form , if the Möbius scalar curvature of the immersion is constant and the Möbius sectional curvature of the immersion satisfies when and when . Then, is Möbius equivalent to either the Riemannian product , for , in ; or is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in .
J. Math. Soc. Japan, Volume 58, Number 3 (2006), 903-925.
First available in Project Euclid: 23 August 2006
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Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
CHENG, Qing-Ming; SHU, Shichang. A Möbius characterization of submanifolds. J. Math. Soc. Japan 58 (2006), no. 3, 903--925. doi:10.2969/jmsj/1156342043. https://projecteuclid.org/euclid.jmsj/1156342043