Second fundamental form

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:

${\displaystyle z=L{\frac {x^{2}}{2}}+Mxy+N{\frac {y^{2}}{2}}+{\text{higher order terms}}\,,}$

and the second fundamental form at the origin in the coordinates (x,y) is the quadratic form

${\displaystyle L\,dx^{2}+2M\,dx\,dy+N\,dy^{2}\,.}$

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 r_u and r_v. Regularity of the parametrization means that r_u and r_v 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 r_u × r_v is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

${\displaystyle \mathbf {n} ={\frac {\mathbf {r} _{u}\times \mathbf {r} _{v}}{|\mathbf {r} _{u}\times \mathbf {r} _{v}|}}\,.}$

The second fundamental form is usually written as

${\displaystyle \mathrm {I\!I} =L\,du^{2}+2M\,du\,dv+N\,dv^{2}\,,}$

its matrix in the basis {r_u, r_v} of the tangent plane is

${\displaystyle {\begin{bmatrix}L&M\\M&N\end{bmatrix}}\,.}$

The coefficients L, M, N at a given point in the parametric uv-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed with the aid of the dot product as follows:

${\displaystyle L=\mathbf {r} _{uu}\cdot \mathbf {n} \,,\quad M=\mathbf {r} _{uv}\cdot \mathbf {n} \,,\quad N=\mathbf {r} _{vv}\cdot \mathbf {n} \,.}$

For a signed distance field of Hessian H, the second fundamental form coefficients can be computed as follows:

${\displaystyle L=-\mathbf {r} _{u}\cdot \mathbf {H} \cdot \mathbf {r} _{u}\,,\quad M=-\mathbf {r} _{u}\cdot \mathbf {H} \cdot \mathbf {r} _{v}\,,\quad N=-\mathbf {r} _{v}\cdot \mathbf {H} \cdot \mathbf {r} _{v}\,.}$

The second fundamental form of a general parametric surface S is defined as follows.

Let r = r(u¹,u²) 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^α by r_α, α = 1, 2. Regularity of the parametrization means that r₁ and r₂ are linearly independent for any (u¹,u²) in the domain of r, and hence span the tangent plane to S at each point. Equivalently, the cross product r₁ × r₂ is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors n:

${\displaystyle \mathbf {n} ={\frac {\mathbf {r} _{1}\times \mathbf {r} _{2}}{|\mathbf {r} _{1}\times \mathbf {r} _{2}|}}\,.}$

The second fundamental form is usually written as

${\displaystyle \mathrm {I\!I} =b_{\alpha \beta }\,du^{\alpha }\,du^{\beta }\,.}$

The equation above uses the Einstein summation convention.

The coefficients b_αβ at a given point in the parametric u¹u²-plane are given by the projections of the second partial derivatives of r at that point onto the normal line to S and can be computed in terms of the normal vector n as follows:

${\displaystyle b_{\alpha \beta }=r_{\,\alpha \beta }^{\ \ \,\gamma }n_{\gamma }\,.}$

The second fundamental form can be generalized to arbitrary codimension. In that case it is a quadratic form on the tangent space with values in the normal bundle and it can be defined by

${\displaystyle \mathrm {I\!I} (v,w)=(\nabla _{v}w)^{\bot }\,,}$

where (∇_vw)^⊥ denotes the orthogonal projection of covariant derivative ∇_vw onto the normal bundle.

In Euclidean space, the curvature tensor of a submanifold can be described by the following formula:

${\displaystyle \langle R(u,v)w,z\rangle =\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle .}$

This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium.

For general Riemannian manifolds one has to add the curvature of ambient space; if N is a manifold embedded in a Riemannian manifold (M,g) then the curvature tensor R_N of N with induced metric can be expressed using the second fundamental form and R_M, the curvature tensor of M:

${\displaystyle \langle R_{N}(u,v)w,z\rangle =\langle R_{M}(u,v)w,z\rangle +\langle \mathrm {I} \!\mathrm {I} (u,z),\mathrm {I} \!\mathrm {I} (v,w)\rangle -\langle \mathrm {I} \!\mathrm {I} (u,w),\mathrm {I} \!\mathrm {I} (v,z)\rangle \,.}$