Second fundamental form

Short description: Quadratic form related to curvatures of surfaces

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 [math]\displaystyle{ \mathrm{I\!I} }[/math] (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

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:

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

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

[math]\displaystyle{ L \, dx^2 + 2M \, dx \, dy + N \, dy^2 \,. }[/math]

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:

[math]\displaystyle{ \mathbf{n} = \frac{\mathbf{r}_u\times\mathbf{r}_v}{|\mathbf{r}_u\times\mathbf{r}_v|} \,. }[/math]

The second fundamental form is usually written as

[math]\displaystyle{ \mathrm{I\!I} = L\, du^2 + 2M\, du\, dv + N\, dv^2 \,, }[/math]

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

[math]\displaystyle{ \begin{bmatrix} L&M\\ M&N \end{bmatrix} \,. }[/math]

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:

[math]\displaystyle{ L = \mathbf{r}_{uu} \cdot \mathbf{n}\,, \quad M = \mathbf{r}_{uv} \cdot \mathbf{n}\,, \quad N = \mathbf{r}_{vv} \cdot \mathbf{n}\,. }[/math]

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

[math]\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\,. }[/math]

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:

[math]\displaystyle{ \mathbf{n} = \frac{\mathbf{r}_1\times\mathbf{r}_2}{|\mathbf{r}_1\times\mathbf{r}_2|}\,. }[/math]

The second fundamental form is usually written as

[math]\displaystyle{ \mathrm{I\!I} = b_{\alpha \beta} \, du^{\alpha} \, du^{\beta} \,. }[/math]

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:

[math]\displaystyle{ b_{\alpha \beta} = r_{,\alpha \beta}^{\ \ \,\gamma} n_{\gamma}\,. }[/math]

Hypersurface in a Riemannian manifold

In Euclidean space, the second fundamental form is given by

[math]\displaystyle{ \mathrm{I\!I}(v,w) = -\langle d\nu(v),w\rangle\nu }[/math]

where [math]\displaystyle{ \nu }[/math] is the Gauss map, and [math]\displaystyle{ d\nu }[/math] the differential of [math]\displaystyle{ \nu }[/math] 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,

[math]\displaystyle{ \mathrm I\!\mathrm I(v,w)=\langle S(v),w\rangle n = -\langle \nabla_v n,w\rangle n=\langle n,\nabla_v w\rangle n\,, }[/math]

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

[math]\displaystyle{ \mathrm{I\!I}(v,w)=(\nabla_v w)^\bot\,, }[/math]

where [math]\displaystyle{ (\nabla_v w)^\bot }[/math] denotes the orthogonal projection of covariant derivative [math]\displaystyle{ \nabla_v w }[/math] onto the normal bundle.

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

[math]\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. }[/math]

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:

[math]\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\,. }[/math]

See also

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

References

• Guggenheimer, Heinrich (1977). "Chapter 10. Surfaces". Differential Geometry. Dover. ISBN 0-486-63433-7. 
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 2 (New ed.). Wiley-Interscience. ISBN 0-471-15732-5. 
• Spivak, Michael (1999). A Comprehensive introduction to differential geometry (Volume 3). Publish or Perish. ISBN 0-914098-72-1. 

External links

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