New Results on Derivatives of the Shape Operator of Real Hypersurfaces in the Complex Quadric

Year 2024, Volume: 17 Issue: 1, 221 - 231, 23.04.2024

Juan De Dios Perez , David Pérez-lópez

https://doi.org/10.36890/iejg.1466325

Abstract

A real hypersurface $M$ in the complex quadric $Q^{m}=SO_{m+2}/SO_mSO_2$ inherits an almost contact metric structure . This structure allows to define, for any nonnull real number $k$, the so called $k$-th generalized Tanaka-Webster connection on $M$, $\hat{\nabla}^{(k)}$. If $\nabla$ denotes the Levi-Civita connection on $M$, we introduce the concepts of $(\hat{\nabla}^{(k)},\nabla)$-Codazzi and $(\hat{\nabla}^{(k)},\nabla)$-Killing shape operator $S$ of the real hypersurface and classify real hypersurfaces in $Q$ satisfying any of these conditions.

Keywords

Complex quadric, real hypersurface, shape operator, $k$-th generalized Tanaka-Webster connection, Cho operators

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