N-dimensional totally real minimal submanifolds of CP n

Abstract

Let CPn be the n-dimensional complex projective space with the Study-Fubini metric of constant holomorphic sectional curvature 4 and let M be a compact, orientable, n-dimensional totally real minimal submanifold of CPn. In this paper we prove the following results. (a) If M is 6-dimensional, conformally flat and has non negative Euler number and constant scalar curvature τ, 0<τ ≦ 70/3, then M is locally isometric to S1,5 :=S1 (sin θ cos θ) × S5 (sin θ), tan θ = √6. (b) If M is 4-dimensional, has parallel second fundamental form and scalar curvature τ ≧ 15/2, then M is locally isometric to S1,3 :=S1 (sin θ cos θ) × S3 (sinθ), tan θ=2, or it is totally geodesic.