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

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 $S{L}_n(\mathbb{C})/SU(n),$ 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 $S{L}_n(\mathbb{C})/SU(n).$ As a new application, we obtain a fundamental theorem for the submanifold theory in that spaces.

34 pages, to appear in Advances in Applied Clifford Algebras (this version of the paper contains many improvements with respect to the first submission)