Flat totally real submanifolds of \(CP^ n\) and the symmetric generalized wave equation (Q1893806)

It is shown that the Hopf fibration \(\pi:S^{2n+1}(1)\to\mathbb{C} P^n\) establishes a one-to-one correspondence between the set of symmetric flat \((n+1)\)-dimensional submanifolds of the unit sphere \(S^{2n+1}(1)\) and the set of \(n\)-dimensional flat totally real submanifolds of the complex projective space \(\mathbb{C} P^n\). It is also proved that a complete flat totally real \(n\)-dimensional submanifold of \(\mathbb{C} P^n\) with mean curvature of constant length is a flat torus.