Equivariant totally real 3-spheres in the complex projective space CPnCPn

In this paper we study the equivariant totally real immersions from S3S3 into CPnCPn. We first reduce these immersions to a system of algebraic equations by the unitary representations of SU(2)SU(2). We give some explicit examples of minimal totally real isometric immersions from S3/m(m+2)3 into CPnCPn, and characterize the minimal totally real isometric immersions from S3/m(m+2)3 into CPnCPn by the standard example. We also give many minimal linearly full isometric immersions from S1/53 into CP7CP7, CP11CP11 and CP15CP15. As an application of our method, we classify equivariant Lagrangian S3S3 in CP3CP3 again.