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Let  X  be the quotient of an irreducible bounded symmetric domain Ω by a lattice. In order to characterize algebraic correspondences on  X  commuting with exterior Hecke correspondences, Clozel–Ullmo studied certain germs of measure-preserving maps from (Ω; 0) into its Cartesian products, proving that such maps are totally geodesic when dim( X ) = 1. Here we prove total geodesy when dim(Ω) ≧ 2 by methods of analytic continuation. For  B   n  ,  n  ≧ 2, total geodesy follows then from Alexander's theorem. When rank(Ω) ≧ 2, we deduce total geodesy from Alexander-type theorems, especially from a new Alexander-type theorem involving Reg(∂Ω) in place of the Shilov boundary.

Germs of measure-preserving holomorphic maps from bounded symmetric domains to their Cartesian products