$H$-umbilical Lagrangian submanifolds of the nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$

TL;DR

This paper proves that in the nearly K"ahler manifold $ ext{S}^3 imes ext{S}^3$, $H$-umbilical Lagrangian submanifolds are necessarily totally geodesic, extending the understanding of their geometric properties.

Contribution

It establishes that $H$-umbilical Lagrangian submanifolds in the homogeneous nearly K"ahler $ ext{S}^3 imes ext{S}^3$ are automatically totally geodesic, a new result in differential geometry.

Findings

$H$-umbilical Lagrangian submanifolds are totally geodesic in $ ext{S}^3 imes ext{S}^3$

Extension of total umbilicity concepts to nearly K"ahler manifolds

New insights into the geometry of Lagrangian submanifolds

Abstract

$H$-umbilicity was introduced as an analogue of total umbilicity for Lagrangian submanifolds since, in some relevant cases, totally umbilical Lagrangian submanifolds are automatically totally geodesic. In this paper, we show that in the homogeneous nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$, also $H$-umbilical Lagrangian submanifolds are automatically totally geodesic.