\(n\)-dimensional totally real minimal submanifolds of \(CP^ n\)
\(S_{1,n-1}=S^1(\sin\theta \cos\theta)\times S^{n-1}(\sin\theta)\), \(\tan\theta=\sqrt n\), can be immersed as a totally real minimal submanifold in \(P_n(\mathbb{C})\) with parallel second fundamental form. The author proves that, among all 6-dimensional conformally flat minimal submanifolds in \(P_6(\mathbb{C})\), \(S_{1,5}\) is characterized as the one with positive Euler number and constant scalar curvature \(\tau\), \(0<\tau\leq 70/3\). A similar result was obtained by \textit{N. Ejiri} [Proc. Am. Math. Soc. 86, 496-497 (1982; Zbl 0499.53046)] when \(n=4\). The author also proves that, among all 4-dimensional totally real minimal submanifolds in \(P_4(\mathbb{C})\) with parallel second fundamental form, \(S_{1,3}\) can be characterized as the one with scalar curvature \(\geq 13/2\).
scientific article; zbMATH DE number 1028205
scientific article