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 <math>\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

File:Second fundamental form.svg

Definition of second fundamental form

Motivation

The second fundamental form of a parametric surface Template:Math in Template:Math was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, Template:Math, and that the plane Template:Math is tangent to the surface at the origin. Then Template:Math and its partial derivatives with respect to Template:Math and Template:Math vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:

<math> 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 Template:Math is the quadratic form

<math> L \, dx^2 + 2M \, dx \, dy + N \, dy^2 \,. </math>

For a smooth point Template:Math on Template:Math, one can choose the coordinate system so that the plane Template:Math is tangent to Template:Math at Template:Math, 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 Template:Math be a regular parametrization of a surface in Template:Math, where Template:Math is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of Template:Math with respect to Template:Math and Template:Math by Template:Math and Template:Math. Regularity of the parametrization means that Template:Math and Template:Math are linearly independent for any Template:Math in the domain of Template:Math, and hence span the tangent plane to Template:Math at each point. Equivalently, the cross product Template:Math is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors Template:Math:

<math>\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>\mathrm{I\!I} = L\, du^2 + 2M\, du\, dv + N\, dv^2 \,,</math>

its matrix in the basis Template:Math of the tangent plane is

<math> \begin{bmatrix}

L&M\\ M&N \end{bmatrix} \,. </math>

The coefficients Template:Math at a given point in the parametric Template:Math-plane are given by the projections of the second partial derivatives of Template:Math at that point onto the normal line to Template:Math and can be computed with the aid of the dot product as follows:

<math>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 Template:Math, the second fundamental form coefficients can be computed as follows:

<math>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 Template:Math is defined as follows.

Let Template:Math be a regular parametrization of a surface in Template:Math, where Template:Math is a smooth vector-valued function of two variables. It is common to denote the partial derivatives of Template:Math with respect to Template:Math by Template:Math, Template:Math. Regularity of the parametrization means that Template:Math and Template:Math are linearly independent for any Template:Math in the domain of Template:Math, and hence span the tangent plane to Template:Math at each point. Equivalently, the cross product Template:Math is a nonzero vector normal to the surface. The parametrization thus defines a field of unit normal vectors Template:Math:

<math>\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>\mathrm{I\!I} = b_{\alpha \beta} \, du^{\alpha} \, du^{\beta} \,.</math>

The equation above uses the Einstein summation convention.

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

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

where <math>\nu</math> is the Gauss map, and <math>d\nu</math> the differential of <math>\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 Template:Math) of a hypersurface,

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

where Template:Math denotes the covariant derivative of the ambient manifold and Template:Math 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 Template:Math (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>\mathrm{I\!I}(v,w)=(\nabla_v w)^\bot\,, </math>

where <math>(\nabla_v w)^\bot</math> denotes the orthogonal projection of covariant derivative <math>\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>\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).</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 Template:Math is a manifold embedded in a Riemannian manifold Template:Math then the curvature tensor Template:Math of Template:Math with induced metric can be expressed using the second fundamental form and Template:Math, the curvature tensor of Template:Math:

<math>\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

• Template:Cite book
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External links

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