Lagrangian warped product immersions in $\mathbb{S}^6$

TL;DR

This paper classifies Lagrangian warped product immersions in the nearly Kähler 6-sphere, identifying conditions under which they are totally geodesic, have constant curvature, or satisfy Chen's inequality with explicit forms.

Contribution

It provides a complete classification of Lagrangian warped product immersions in $S^6$, including explicit descriptions when they satisfy Chen's inequality.

Findings

Totally geodesic Lagrangian immersions identified.

Immersions with constant sectional curvature 1/16 characterized.

Explicit forms given for immersions satisfying Chen's inequality.

Abstract

We study Lagrangian immersions in the nearly K\"ahler $\mathbb{S}^6$ which are warped product manifolds of a $1$-dimensional base and a surface. Apart from the totally geodesic ones, they are either of constant sectional curvature $\frac{1}{16}$ or they satisfy equality in Chen's inequality, in which case the immersion is given explicitly.