FLAT TOTALLY REAL SUBMANIFOLDS OF CPⁿ AND THE SYMMETRIC GENERALIZED WAVE EQUATION

Abstract

In this paper we show that the projection of the Hopf fibration establishes a one-to-one correspondence between the set of symmetric flat submanifolds in Euclidean sphere and the set of totally real flat submanifolds in complex projective space with the same codimension. We also show that any complete totally real submanifold in complex projective space with mean curvature of constant length and equal dimension and codimension is a flat torus.