Spinorial representation of submanifolds in metric Lie groups

TL;DR

This paper develops a spinorial framework to represent submanifolds within metric Lie groups, providing new proofs, recovering known representations, and extending to surfaces in homogeneous spaces and CMC-surfaces.

Contribution

It introduces a comprehensive spinorial representation for submanifolds in metric Lie groups, unifies existing models, and extends to new classes of surfaces.

Findings

Provides a spinorial proof of the Fundamental Theorem for submanifolds in Lie groups

Recovers known representations in Euclidean space and specific Lie groups

Introduces a new spinorial representation for surfaces in 3D semi-direct product spaces

Abstract

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Lie groups equipped with left invariant metrics. As applications, we get a spinorial proof of the Fundamental Theorem for submanifolds into Lie groups, we recover previously known representations of submanifolds in $\mathbb{R}^n$ and in the 3-dimensional Lie groups $S^3$ and $E(\kappa,\tau),$ and we get a new spinorial representation for surfaces in the 3-dimensional semi-direct products: this achieves the spinorial representations of surfaces in the 3-dimensional homogeneous spaces. We finally indicate how to recover a Weierstrass-type representation for CMC-surfaces in 3-dimensional metric Lie groups recently given by Meeks, Mira, Perez and Ros.