Rigidity of the almost complex surfaces in the nearly Kähler S3×S3S3×S3

In this paper we first show that the well-known nearly Kähler manifold S3×S3S3×S3 is neither locally symmetric nor Chern flat. Then, by studying the rigidity of compact almost complex surfaces in S3×S3S3×S3, we establish a Simons’ type integral inequality so that we obtain a new characterization of two typical examples of totally geodesic almost complex surfaces in S3×S3S3×S3.