Bonnet's theorem in geometry

Bonnet's theorem states that if two surfaces have identical intrinsic properties—meaning measurements like distances and angles on their surfaces are the same—they must be basically the same shape, even if their actual forms look different. In other words, the theorem links the surface's internal geometry directly to its shape in space. It emphasizes that knowing how a surface is curved and measured locally determines its overall shape, up to rigid motions like bending or twisting without stretching. This principle is fundamental in differential geometry, connecting local surface properties to global forms.