Scalar curvature and real submanifolds of codimension 2 of a complex projective space
One of the main problems concerning real submanifolds of a complex manifold is to give nice conditions for the submanifold to be a complex submanifold. In the case that the ambient manifold is a complex projective space H. B. Lawson gave a condition for an even-dimensional real minimal submanifold to be not only a complex submanifold but also a totally geodesic complex linear subspace. The condition is given by an inequality between the scalar curvature of the submanifold and that of the ambient manifold. In this paper we study real submanifolds of codimension 2 that are not necessarily minimal and prove that under a suitable condition on the normal connection, if the scalar curvature of the submanifold is not smaller than that of a standard complex linar subspace, then the submanifold is necessarily a totally geodesic complex linear subspace.
scientific article; zbMATH DE number 3897777