Fundamentformen

Fundamentformen, or fundamental forms, is a term used in differential geometry to describe two quadratic forms associated with a smooth surface embedded in three-dimensional space. They encode metric and curvatural information about the surface and are central to the classical theory of surfaces.

The first fundamental form, I, captures metric properties intrinsic to the surface. For a parametrization r(u,v) of a surface, it is written as ds^2 = E du^2 + 2F du dv + G dv^2, where E = r_u · r_u, F = r_u · r_v, and G = r_v · r_v. This form determines lengths of tangent vectors, angles between curves, and area elements on the surface via the area element dA = sqrt(EG − F^2) du dv.

The second fundamental form, II, encodes how the surface bends in space. With unit normal n, it is II = e du^2 + 2f du dv + g dv^2, where e = n · r_uu, f = n · r_uv, and g = n · r_vv. The second form relates to curvature through the shape operator S, satisfying II(X,Y) = ⟨S X, Y⟩. The principal curvatures k1 and k2 are eigenvalues of S, and Gaussian and mean curvatures follow as K = (eg − f^2)/(EG − F^2) and H = (E g − 2F f + G e)/(2(EG − F^2)).

The fundamental forms are connected by Gauss-Codazzi equations, which are the compatibility conditions ensuring the existence (locally) of a surface in space with given I and II up to rigid motion. The term is widely used in German-language literature as the plural fundamentformen.