Real hypersurfaces in the complex quadric with Reeb parallel structure Jacobi operator

Abstract

In this paper, we first introduce the full express of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^m$ from the equation of Gauss. Next we derive a formula for the structure Jacobi operator $R_{\xi }$ and its derivative under the Levi-Civita connection of $M$. We give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, ${\nabla }_{\xi }{R}_{\xi }=0$, in the complex quadric $Q^m$, $m\ge 3$.

14 pages. arXiv admin note: substantial text overlap with arXiv:1605.05316, arXiv:1512.03671; text overlap with arXiv:1907.03374; text overlap with arXiv:1710.10627 by other authors