Cartan's Theorem

Cartan's Theorem states that for certain types of geometric structures called holomorphic fiber bundles, every continuous symmetry (or automorphism) can be realized as an algebraic transformation. In simpler terms, it means that the symmetries of these complex geometric objects are not just continuous or smooth, but have a precise algebraic description. This result helps mathematicians understand how these geometric shapes can be transformed and classified, ensuring that their symmetries are deeply connected to their algebraic structure, which is fundamental for areas like complex geometry and differential equations.