Second fundamental form

Template:Short description 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 $\mathrm{I}\mathrm{I}$ (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.

Surface in R³

File:Second fundamental form.svg

Definition of second fundamental form

Motivation

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:

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

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

$Ld{x}^2+2Mdxdy+Nd{y}^2.$

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.

Classical notation

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:

$\mathbf{𝐧}=\frac{{\mathbf{𝐫}}_u\times {\mathbf{𝐫}}_v}{|{\mathbf{𝐫}}_u\times {\mathbf{𝐫}}_v|}.$

The second fundamental form is usually written as

$\mathrm{I}\mathrm{I}=Ld{u}^2+2Mdudv+Nd{v}^2,$

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

$\left[\begin{array}{cc}L& M\\ M& N\end{array}\right].$

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:

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

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

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

Physicist's notation

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:

$\mathbf{𝐧}=\frac{{\mathbf{𝐫}}_1\times {\mathbf{𝐫}}_2}{|{\mathbf{𝐫}}_1\times {\mathbf{𝐫}}_2|}.$

The second fundamental form is usually written as

$\mathrm{I}\mathrm{I}=b_{\alpha \beta }d{u}^{\alpha }d{u}^{\beta }.$

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:

$b_{\alpha \beta }=r_{,\alpha \beta }^{\gamma }{n}_{\gamma }.$

Hypersurface in a Riemannian manifold

In Euclidean space, the second fundamental form is given by

$\mathrm{I}\mathrm{I}(v,w)=-⟨d\nu (v),w⟩\nu$

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,

$\mathrm{I}\mathrm{I}(v,w)=⟨S(v),w⟩=-⟨{\mathrm{\nabla }}_{v}n,w⟩=⟨n,{\mathrm{\nabla }}_{v}w⟩,$

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).

Generalization to arbitrary codimension

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

$\mathrm{I}\mathrm{I}(v,w)=({\mathrm{\nabla }}_{v}w{)}^{\mathrm{\perp }},$

where $({\mathrm{\nabla }}_{v}w{)}^{\mathrm{\perp }}$ denotes the orthogonal projection of covariant derivative ${\mathrm{\nabla }}_{v}w$ onto the normal bundle.

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

$⟨R(u,v)w,z⟩=\mathrm{I}\mathrm{I}(u,z)\mathrm{I}\mathrm{I}(v,w)-\mathrm{I}\mathrm{I}(u,w)\mathrm{I}\mathrm{I}(v,z).$

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:

$⟨R_N(u,v)w,z⟩=⟨R_M(u,v)w,z⟩+⟨\mathrm{I}\mathrm{I}(u,z),\mathrm{I}\mathrm{I}(v,w)⟩-⟨\mathrm{I}\mathrm{I}(u,w),\mathrm{I}\mathrm{I}(v,z)⟩.$

See also

• First fundamental form
• Gaussian curvature
• Gauss–Codazzi equations
• Shape operator
• Third fundamental form
• Tautological one-form

External links

• Steven Verpoort (2008) Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects from Katholieke Universiteit Leuven.