Spinorial representation of submanifolds in $SL_n(\mathbb{C})/SU(n)$

TL;DR

This paper develops a spinorial representation framework for submanifolds in symmetric spaces like $SL_n(\mathbb{C})/SU(n)$, generalizing known surface representations and providing a new fundamental theorem for submanifold theory.

Contribution

It introduces a general spinorial representation for submanifolds in symmetric spaces, extending previous surface theories to higher dimensions and co-dimensions.

Findings

Recovered Bryant's representation for CMC-1 surfaces in hyperbolic space

Extended spinorial representation to higher-dimensional symmetric spaces

Established a fundamental theorem for submanifold theory in these spaces

Abstract

We give a spinorial representation of a submanifold of any dimension and co-dimension in a symmetric space $G/H,$ where $G$ is a complex semi-simple Lie group and $H$ is a compact real form of $G.$ This in particular includes $SL_n(\mathbb{C})/SU(n),$ and extends the previously known spinorial representation of a surface in $\mathbb{H}^3$ if $n=2.$ We also recover the Bryant representation of a surface with constant mean curvature 1 in $\mathbb{H}^3$ and its generalization for a surface with holomorphic right Gauss map in $SL_n(\mathbb{C})/SU(n).$ As a new application, we obtain a fundamental theorem for the submanifold theory in that spaces.