×

Ricci generalized pseudo-parallel Kählerian submanifolds in complex space forms. (English) Zbl 1167.53025

The authors study Kählerian submanifolds \(M\) of complex dimension \(n\) in a complex \(m\)–dimensional space form of constant holomorphic sectional curvature \(c\). These submanifolds are assumed to be Ricci generalized pseudo–parallel. This is a weakening of the condition that the second fundamental form \(h\) is parallel. Namely, one assumes that the action of the curvature on \(h\) can be expressed in a certain way by the Ricci tensor, \(h\) itself and an auxiliary smooth function \(L\). Denoting by \(\tau\) the scalar curvature of \(M\), the main theorem of the article states that either \(M\) is totally geodesic, or the smooth function \(\|h\|^2+\frac{2}{3}(L\tau-\frac{1}{2}(n+2)c)\) is either identically zero or strictly positive in one point. The authors derive several corollaries, in particular various sufficient conditions for \(M\) being totally geodesic.
Reviewer: Andreas Cap (Wien)

MSC:

53B25 Local submanifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
PDFBibTeX XMLCite
Full Text: EuDML Link