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 hypersurface in a Riemannian manifold and a smooth choice of the unit normal vector at each point.

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:

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

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

$L \, \text{d}x^2 + 2M \, \text{d}x \, \text{d}y + N \, \text{d}y^2. \,$

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.

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

$\mathrm{I\!I} = L\, \text{d}u^2 + 2M\, \text{d}u\, \text{d}v + N\, \text{d}v^2, \,$

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:

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

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

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

The equation above uses the Einstein Summation Convention. The coefficients b_αβ at a given point in the parametric (u¹, u²)-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

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

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,

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

where $\nabla_v w$ 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

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

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:

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

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 $M$:

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

See also

• First fundamental form
• Gaussian curvature
• Gauss–Codazzi equations
• Shape operator

External links

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