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

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.