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 .
Qing-Ming CHENG. Shichang SHU. "A Möbius characterization of submanifolds." J. Math. Soc. Japan 58 (3) 903 - 925, July, 2006. https://doi.org/10.2969/jmsj/1156342043