On the scalar curvature and sectional curvatures of a Kaehler submanifold

Abstract

For a Kaehler submanifold of a complex space form, pinching for scalar curvature implies pinching for sectional curvatures. 1 . Statement of result. The scalar curvature is, by definition, the sum of Ricci curvatures with respect to an orthonormal basis of the tangent space, and the Ricci curvature is the sum of sectional curvatures. Therefore, in general, it may be very difficult to expect some implications from the scalar curvature to the sectional curvature. However, we shall show in this note that for a Kaehler submanifold of a complex space form, pinching for scalar curvature implies pinching for sectional curvatures. Precisely, we shall prove the following pointwise theorem: THEOREM.Let M be an n-dimensional Kaehler submanifold of an (n+p)-dimensional Kaehler manifold of constant holomorphic sectional curuature c. If the scalar curvature p of M satisfies p Z n ( n f 1 ) c a at a point P , then every anti-holomorphic sectional curvature of M at P is z k ( c a ) . Although we may expect some better implication under some additional assumptions (for example, the compactness of M), our theorem is proved without any global assumption. REMARK1. Let llol] be the length of the second fundamental form o of the immersion. Then we have p=n(n+ 1)cjlo1I2so that the assumption pzn(n+l)c-a is equivalent to llo112sa. REMARK2. Our theorem can be considered as a complex version of Theorem 1 of [I], in the proof of which we find a minor mistake. The :orrection will be given in the Appendix. 2. Proof of theorem. Let @,,,(c) be an (n+p)-dimensional Kaehler manifold of constant holomorphic sectional curvature c, and let M be Received by the editors February 26, 1973. A M S ( M O S ) subject classififations (1970). Primary 53B25, 53B35.