A Möbius characterization of submanifolds

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$¥mapsto$S^n+p(1) without umbilical points and with vanishing Möbius form Φ, if (n-2)||Ã||≤$¥sqrt{¥smash[b]{¥frac{n-1}{n}}} ¥big¥{ nR-¥frac{1}{n}[(n-1)¥big( 2-¥frac{1}{p} ¥big)-1] ¥big¥}$ is satisfied, then, x is Möbius equivalent to an open part of either the Riemannian product S^n-1(r)×S¹($¥sqrt{1-r^2}$) in Sⁿ⁺¹(1), or the image of the conformal diffeomorphism σ of the standard cylinder S^n-1(1)×R in Rⁿ⁺¹, or the image of the conformal diffeomorphism τ of the Riemannian product S^n-1(r)×H¹($¥sqrt{1+r^2}$) in Hⁿ⁺¹, or x is locally Möbius equivalent to the Veronese surface in S⁴(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$¥mapsto$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($¥sqrt{1-r^2}$), for k=1, 2, …, n-1, in Sⁿ⁺¹(1); or x is Möbius equivalent to a compact minimal submanifold with constant scalar curvature in S^n+p(1).