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

## TL;DR

This paper classifies Hopf real hypersurfaces in complex quadrics with Reeb parallel structure Jacobi operator, providing explicit curvature tensor formulas and a complete classification for dimensions m ≥ 3.

## Contribution

It introduces explicit formulas for the curvature tensor and Jacobi operator, and fully classifies Hopf hypersurfaces with Reeb parallel structure Jacobi operator in complex quadrics.

## Key findings

- Complete classification of Hopf hypersurfaces with Reeb parallel Jacobi operator in Q^m for m ≥ 3.
- Explicit formulas for the Riemannian curvature tensor of hypersurfaces in complex quadrics.
- Derivation of the structure Jacobi operator and its covariant derivative.

## 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 $Q^{m}$, $m \geq 3$.