Journal of the Mathematical Society of Japan

A Möbius characterization of submanifolds

Qing-Ming CHENG and Shichang SHU

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Abstract

In this paper, we study Möbius characterizations of submanifolds without umbilical points in a unit sphere S n + p ( 1 ) . First of all, we proved that, for an n -dimensional ( n 2 ) submanifold x : M S n + p ( 1 ) without umbilical points and with vanishing Möbius form Φ , if ( n - 2 ) | | A ˜ | | n - 1 n n R - 1 n [ ( n - 1 ) 2 - 1 p - 1 ] is satisfied, then, x is Möbius equivalent to an open part of either the Riemannian product S n - 1 ( r ) × S 1 1 - r 2 in S n + 1 ( 1 ) , or the image of the conformal diffeomorphism σ of the standard cylinder S n - 1 ( 1 ) × R in R n + 1 , or the image of the conformal diffeomorphism τ of the Riemannian product S n - 1 ( r ) × H 1 1 + r 2 in H n + 1 , or x is locally Möbius equivalent to the Veronese surface in S 4 ( 1 ) . When p = 1 , our pinching condition is the same as in Main Theorem of Hu and Li [6], in which they assumed that M is compact and the Möbius scalar curvature n ( n - 1 ) R is constant. Secondly, we consider the Möbius sectional curvature of the immersion x . We obtained that, for an n -dimensional compact submanifold x : M S n + p ( 1 ) without umbilical points and with vanishing form Φ , if the Möbius scalar curvature n ( n - 1 ) R of the immersion x is constant and the Möbius sectional curvature K of the immersion x satisfies K 0 when p = 1 and K > 0 when p > 1 . Then, x is Möbius equivalent to either the Riemannian product S k ( r ) × S n - k 1 - r 2 , for k = 1 , 2 , , n - 1 , in S n + 1 ( 1 ) ; or x is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in S n + p ( 1 ) .

Article information

Source
J. Math. Soc. Japan, Volume 58, Number 3 (2006), 903-925.

Dates
First available in Project Euclid: 23 August 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1156342043

Digital Object Identifier
doi:10.2969/jmsj/1156342043

Mathematical Reviews number (MathSciNet)
MR2254416

Zentralblatt MATH identifier
1102.53009

Subjects
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]

Keywords
submanifold Möbius metric Möbius scalar curvature Möbius sectional curvature Blaschke tensor and Möbius form

Citation

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


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