Communications of the
Korean Mathematical Society CKMS

In this paper we prove the following : Let {\it M} be a real (2n-1)-dimensional compact minimal semi-invariant submanifold in a complex projective space $P_{n+1} C$. If the scalar curvature $\geq 2(n-1)(2n+1)$, then {\it M} is a homogeneous type A$_1$ or A$_2$. Next suppose that the third fundamental form $n$ satisfies $dn=2\theta \omega$ for a certain scalar $\theta \ne {c \over 2}$ and $\theta \ne {c \over 4} {4n-1 \over 2n-1}$, where $\omega(X,Y)=g(X,\phi Y)$ for any vectors {\it X} and {\it Y} on a semi-invariant submanifold of codimension 3 in a complex space form $M_{n+1}(c)$. Then we prove that {\it M} has constant principal curvatures corresponding the shape operator in the direction of the distinguished normal and the structure vector $\xi$ is an eigenvector of $A$ if and only if {\it M} is locally congruent to a homogeneous minimal real hypersurface of $M_{n} (c)$.

Keywords: semi-invariant minimal submanifold, distinguished normal, homogeneous real hypersurface

MSC numbers: 53C15, 53C25, 53C40