N-DIMENSIONAL TOTALLY-REAL MINIMAL SUBMANIFOLDS OF CPN

Let CPn be the n-dimensional complex projective space with the Study-F ubini metric of constant holomorphic sectional curvature 4 and let M b e a compact, orientable, n-dimensional totally real minimal submanifol d 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 co nstant scalar curvature tau, 0 < tau less than or equal to 70/3, then M is locally isometric to S-1,S-5 := S-1(sin theta cos theta) x S-5(si n theta), tan theta = root 6. (b) If M is 4-dimensional, has parallel second fundamental form and scalar curvature tau greater than or equal to 15/2, then M is locally isometric to S-1,S-3 := S-1(sin theta cos theta) x S-3 (sin theta), tan theta = 2, or it is totally geodesic.