Real hypersurfaces with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians

TL;DR

This paper proves that real hypersurfaces with pseudo-parallel or semi-parallel normal Jacobi operators do not exist in complex two-plane Grassmannians, answering a previously open question negatively.

Contribution

It establishes the non-existence of such hypersurfaces, extending the understanding of geometric structures in complex Grassmannians.

Findings

No real hypersurface with pseudo-parallel normal Jacobi operator exists in complex two-plane Grassmannians.

There are no real hypersurfaces with semi-parallel or recurrent normal Jacobi operators in these spaces.

The paper resolves an open question from prior research negatively.

Abstract

The objective of the present paper is to prove the non-existence of real hypersurface with pseudo-parallel normal Jacobi operator in complex two-plane Grassmannians. As a corollary, we show that there does not exist any real hypersurface with semi-parallel or recurrent normal Jacobi operator in complex two-plane Grassmannians. This answers a question considered in [Monatsh Math, 172 (2013), 167-178] in negative.