Real hypersurfaces in the complex quadric with commuting Ricci tensor

TL;DR

This paper classifies real hypersurfaces in the complex quadric with commuting Ricci tensor, showing that such tensors imply the normal vector field is either -principal or -isotropic, and provides a complete classification in each case.

Contribution

It introduces the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric and classifies all such hypersurfaces based on this property.

Findings

Normal vector field is either -principal or -isotropic for commuting Ricci tensor.

Complete classification of hypersurfaces with commuting Ricci tensor in the complex quadric.

Provides geometric conditions characterizing these hypersurfaces.

Abstract

We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . It is shown that the commuting Ricci tensor gives that the unit normal vector field $N$ becomes $\frak A$-principal or $\frak A$-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in $Q^m = SO_{m+2}/SO_mSO_2$ with commuting Ricci tensor.