On $C$-totally real submanifolds of $\mathbb{S}^{2n+1}(1)$ with non-negative sectional curvature

Abstract

For all $n \geq 4$, we give a complete classification of the compact $n$-dimensional minimal $C$-totally real submanifolds in the $(2n+1)$-dimensional unit sphere $\mathbb S^{2n+1}(1)$ with non-negative sectional curvature. This generalizes the results of Yamaguchi et al (Proc Amer Math Soc 54: 276-280, 1976) for $n$ = 2 and, Dillen and Vrancken (Math J Okayama Univ 31: 227-242, 1989) for $n$ = 3. Additionally, we show that, as compact minimal $C$-totally real submanifolds, the standard embeddings of the symmetric spaces $\mathrm{SU}(m)/\mathrm{SO}(m)$, $\mathrm{SU}(m)$, $\mathrm{SU}(2m)/\mathrm{Sp}(m)$ for each $m \geq 3$, and $\mathrm{E}_{6}/\mathrm{F}_4$ into $\mathbb S^{2n+1}(1)$ are all Willmore submanifolds, with $n=\frac{1}{2}m(m-1)-1$, $m^2-1$, $2m^2-m-1$ and 26, respectively.