Third fundamental form

The third fundamental form is a concept from differential geometry, particularly in the study of surfaces within three-dimensional Euclidean space (or higher-dimensional spaces). It is related to the intrinsic and extrinsic properties of surfaces. In the context of a surface \( S \) in three-dimensional Euclidean space, the first and second fundamental forms are well-known constructs used to describe the metric properties of the surface. These forms give insights into lengths, angles, and curvatures.