Parallel Kähler submanifolds and 𝑅-spaces

Abstract

Parallel Kähler submanifolds here mean complex submanifolds immersed in complex projective spaces with parallel second fundamental form. The classification of such submanifolds was done first by Hisao Nakagawa and Ryoichi Takagi (1976) and reproduced secondly and thirdly by Masaru Takeuchi (1978, 1984) by two different methods of unitary representation theory and Jordan triple systems. In this article we briefly survey such related submanifold theory and give the fourth proof for the classification theorem by a different approach based on the differential geometric characterization of R R -spaces due to Carlos Olmos and Cristián U. Sánchez (1991).