Unique jet determination and extension of germs of CR maps into spheres

We provide a new way of simultaneously parametrizing arbitrary local CR maps from real-analytic generic manifolds M ⊂ C N M\subset \mathbb {C}^N into spheres alttext="double-struck S 2 N prime minus 1 Baseline prime"> mathvariant="double-struck">S 2 ′ −<!-- − <mml:mn>1 encoding="application/x-tex">\mathbb {S}^{2N’-1}\subset {C}^{N’} any dimension. The parametrization is obtained as composition universal rational with holomorphic map depending only on M"> encoding="application/x-tex">M . As applications, we obtain rigidity results different flavours such unique jet determination and global extension maps.