Second fundamental form

In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by I I {displaystyle mathrm {I!I} } (read 'two'). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point. In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by I I {displaystyle mathrm {I!I} } (read 'two'). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point. The second fundamental form of a parametric surface S in R3 was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms: and the second fundamental form at the origin in the coordinates x, y is the quadratic form For a smooth point P on S, one can choose the coordinate system so that the coordinate z-plane is tangent to S at P and define the second fundamental form in the same way. The second fundamental form of a general parametric surface is defined as follows. Let r = r(u,v) be a regular parametrization of a surface in R3, where r is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of r with respect to u and v by ru and rv. Regularity of the parametrization means that ru and rv are linearly independent for any (u,v) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product ru × rv is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n: