Curvature - Two Dimensions: Curvature of Surfaces - Second Fundamental Form

Second Fundamental Form

The intrinsic and extrinsic curvature of a surface can be combined in the second fundamental form. This is a quadratic form in the tangent plane to the surface at a point whose value at a particular tangent vector X to the surface is the normal component of the acceleration of a curve along the surface tangent to X; that is, it is the normal curvature to a curve tangent to X (see above). Symbolically,

where N is the unit normal to the surface. For unit tangent vectors X, the second fundamental form assumes the maximum value k₁ and minimum value k₂, which occur in the principal directions u₁ and u₂, respectively. Thus, by the principal axis theorem, the second fundamental form is

Thus the second fundamental form encodes both the intrinsic and extrinsic curvatures.

A related notion of curvature is the shape operator, which is a linear operator from the tangent plane to itself. When applied to a tangent vector X to the surface, the shape operator is the tangential component of the rate of change of the normal vector when moved along a curve on the surface tangent to X. The principal curvatures are the eigenvalues of the shape operator, and in fact the shape operator and second fundamental form have the same matrix representation with respect to a pair of orthonormal vectors of the tangent plane. The Gauss curvature is thus the determinant of the shape tensor and the mean curvature is half its trace.