Spinorial representation of submanifolds in metric Lie groups

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.

35 pages, no figures

References in corpus (3)

• Spinorial Representation of Submanifolds in Riemannian Space Forms
• Spinor representation of Lorentzian surfaces in R^{2,2}
• A generalized Weierstrass representation of Lorentzian surfaces in ${\mathbb{R}}^{2,2}$ and applications

Cited by in corpus (2)

• Characterization of hypersurfaces in four dimensional product spaces via two different Spin${}^c$ structures
• Spinorial representation of submanifolds in $S{L}_n(\mathbb{C})/SU(n)$