Hadamard's theorem

Hadamard's theorem states that a smooth, convex shape whose boundary curves outward everywhere—like a perfect, rounded hill—can be flattened (mapped) onto a flat surface without much distortion. Specifically, for certain curved surfaces called "worlds" with no holes and positive curvature everywhere, there's a way to stretch or compress their surface onto a plane so that distances are preserved as much as possible. This theorem ensures such a "nice" mapping exists, which is foundational in areas like differential geometry and complex analysis, helping us understand how curved shapes relate to flat ones.