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 \(\mathrm{I\!I}\) (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 immersed submanifold in a Riemannian manifold.

The second fundamental form of a parametric surface S in R 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:

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

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

\(L \, dx^2 + 2M \, dx \, dy + N \, dy^2 \,.\)

For a smooth point P on S, one can choose the coordinate system so that the plane z = 0 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 R, 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:

\(\mathbf{n} = \frac{\mathbf{r}_u\times\mathbf{r}_v}{|\mathbf{r}_u\times\mathbf{r}_v|} \,.\)

The second fundamental form is usually written as

\(\mathrm{I\!I} = L\, du^2 + 2M\, du\, dv + N\, dv^2 \,;\)

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

\(\begin{bmatrix} L&M\\ M&N \end{bmatrix} \,.\)

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:

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

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

\(L = -\mathbf{r}_u \cdot \mathbf{H} \cdot \mathbf{r}_u\,, \quad M = -\mathbf{r}_u \cdot \mathbf{H} \cdot \mathbf{r}_v\,, \quad N = -\mathbf{r}_v \cdot \mathbf{H} \cdot \mathbf{r}_v\,.\)

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

Let r = r(u,u) be a regular parametrization of a surface in R, 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 by r_α, α = 1, 2. Regularity of the parametrization means that r₁ and r₂ are linearly independent for any (u,u) 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:

\(\mathbf{n} = \frac{\mathbf{r}_1\times\mathbf{r}_2}{|\mathbf{r}_1\times\mathbf{r}_2|}\,.\)

The second fundamental form is usually written as

\(\mathrm{I\!I} = b_{\alpha \beta} \, du^{\alpha} \, du^{\beta} \,.\)

The equation above uses the Einstein summation convention.

The coefficients b_αβ at a given point in the parametric uu-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:

\(b_{\alpha \beta} = r_{,\alpha \beta}^{\ \ \,\gamma} n_{\gamma}\,.\)

In Euclidean space, the second fundamental form is given by

\(\mathrm{I\!I}(v,w) = -\langle d\nu(v),w\rangle\)

where \(\nu\) is the Gauss map, and \(d\nu\) the differential of \(\nu\) regarded as a vector-valued differential form, and the brackets denote the metric tensor of Euclidean space.

More generally, on a Riemannian manifold, the second fundamental form is an equivalent way to describe the shape operator (denoted by S) of a hypersurface,

\(\mathrm I\!\mathrm I(v,w)=\langle S(v),w\rangle = -\langle \nabla_v n,w\rangle=\langle n,\nabla_v w\rangle \,,\)

where ∇_vw denotes the covariant derivative of the ambient manifold and n a field of normal vectors on the hypersurface. (If the affine connection is torsion-free, then the second fundamental form is symmetric.)

The sign of the second fundamental form depends on the choice of direction of n (which is called a co-orientation of the hypersurface - for surfaces in Euclidean space, this is equivalently given by a choice of orientation of the surface).

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

\(\mathrm{I\!I}(v,w)=(\nabla_v w)^\bot\,,\)

where \((\nabla_v w)^\bot\) denotes the orthogonal projection of covariant derivative \(\nabla_v w\) onto the normal bundle.

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

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

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:

\(\langle R_N(u,v)w,z\rangle = \langle R_M(u,v)w,z\rangle+\langle \mathrm I\!\mathrm I(u,z),\mathrm I\!\mathrm I(v,w)\rangle-\langle \mathrm I\!\mathrm I(u,w),\mathrm I\!\mathrm I(v,z)\rangle\,.\)

• The second fundamental form of a plane is identically zero. A plane has a constant unit normal vector \(\mathbf{n}\), so the second derivatives \(\mathbf{r}_{uu}, \mathbf{r}_{uv}, \mathbf{r}_{vv}\) are tangent to the surface. Consequently, all coefficients \(L, M, N\) in the second fundamental form are zero, and \(\mathrm{I\!I} = 0\).

• The second fundamental form of the unit sphere \(x^2 + y^2 + z^2 = 1\) in the local coordinates \(\mathbf{r}(u,v) = (\sin u \cos v, \sin u \sin v, \cos u)\) is \(\mathrm{I\!I} = -du^2 - \sin^2 u dv^2\). The outward unit normal vector \(\mathbf{n}\) coincides with \(\mathbf{r}(u,v)\). Computing dot products of \(\mathbf{n}\) with \(\mathbf{r}_{uu}, \mathbf{r}_{uv}, \mathbf{r}_{vv}\) yields \(L=-1,\, M=0,\, N=-\sin^2 u\).