Derivatives of normal Jacobi operator on real hypersurfaces in the complex quadric

Abstract

In \cite{S 2017}, Suh gave a non-existence theorem for Hopf real hypersurfaces in the complex quadric with parallel normal Jacobi operator. Motivated by this result, in this paper, we introduce some generalized conditions named $\mathcal{C}$-parallel or Reeb parallel normal Jacobi operators. By using such weaker parallelisms of normal Jacobi operator, first we can assert a non-existence theorem of Hopf real hypersurfaces with $\mathcal{C}$-parallel normal Jacobi operator in the complex quadric $Q^m$, $m\ge 3$. Next, we prove that a Hopf real hypersurface has Reeb parallel normal Jacobi operator if and only if it has an $\mathfrak{A}$-isotropic singular normal vector field.

12 pages. arXiv admin note: substantial text overlap with arXiv:1907.04661, arXiv:2003.07231, arXiv:1512.03671, arXiv:1605.05316

References in corpus (1)

• Totally geodesic submanifolds of the complex quadric