Spinorial Representation of Submanifolds in \({\varvec{SL}}_{\varvec{n}}{\varvec{(\mathbb {C})/SU(n)}}\)

  • Pierre BayardEmail author


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.


Spin geometry Isometric immersions Weierstrass representation Symmetric spaces 

Mathematics Subject Classification

53C27 53C35 53C42 



The author is very indebted to the referees for many comments which helped to improve considerably the writing of the paper. The author was supported by the project PAPIIT-UNAM IA106218.


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Authors and Affiliations

  1. 1.Facultad de Ciencias, Universidad Nacional Autónoma de MéxicoCiudad UniversitariaDelegación CoyoacánMexico

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