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

Abstract

For all n ≥ 4, we give a complete classification of the compact n-dimensional minimal C-totally real submanifolds in the (2n + 1)-dimensional unit sphere 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 SU(m)/SO(m), SU(m), SU(2m)/Sp(m) for each m ≥ 3, and E₆/F₄ into are all Willmore submanifolds, with n = 1/2m(m - 1) - 1, m² - 1, 2m² - m - 1 and 26, respectively.