Ricci's Theorem

Ricci’s Theorem states that the way a surface bends in space can be described by its curvature, and this curvature can be understood through relationships between the surface’s geometry. Essentially, it formalizes how the surface's intrinsic properties (like distances and angles on the surface itself) relate to its extrinsic curvature (how it curves in space). The theorem ensures that the intrinsic measurements are consistent with the surface’s shape in three-dimensional space, and it provides the mathematical foundation for understanding how local bending relates to overall shape, fundamental in differential geometry and applications like physics and computer graphics.