Cartan's formalism of moving frames (repère mobile) allows spinors to be defined on curved manifolds via local orthonormal frames (vielbeins) without requiring affine transformation laws.

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Affine transformation laws(as a technical requirement that Cartan's approach avoids needing): Mathematical rules that describe how geometric objects must change when you move or rotate your coordinate system; a stricter requirement than what this method needs.
Cartan(as the mathematician whose generalized spaces are referenced): Élie Cartan was a French mathematician who developed new ways of understanding curved spaces and geometry; his work allows us to describe spaces that bend and twist in complex ways.
Curved manifolds(as the type of space where spinors are being defined): Spaces that are bent or curved (like the surface of a sphere or a saddle) rather than flat, studied in advanced geometry.
Local orthonormal frames (vielbeins)(as the practical tool that makes the moving frames method work): Sets of perpendicular directions at each point in space that help measure distances and angles locally; 'vielbein' is German for 'many legs.'

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