Advances in Applied Clifford Algebras | 2019
Spinorial Representation of Submanifolds in $${\\varvec{SL}}_{\\varvec{n}}{\\varvec{(\\mathbb {C})/SU(n)}}$$SLn(C)/SU(n)
Abstract
We give a spinorial representation of a submanifold of any dimension and co-dimension in a symmetric space G\xa0/\xa0H,\xa0 where G is a complex semi-simple Lie group and H is a compact real form of G. This in particular includes $$SL_n(\\mathbb {C})/SU(n),$$SLn(C)/SU(n), and extends the previously known spinorial representation of a surface in $$\\mathbb {H}^3$$H3 if $$n=2.$$n=2. We also recover the Bryant representation of a surface with constant mean curvature 1 in $$\\mathbb {H}^3$$H3 and its generalization for a surface with holomorphic right Gauss map in $$SL_n(\\mathbb {C})/SU(n).$$SLn(C)/SU(n). As a new application, we obtain a fundamental theorem for the submanifold theory in that spaces.