second fundamental form

In classical differential geometry the second fundamental form is a symmetric bilinear form defined on a differentiable surface $M$ embedded in $\mathbb{R}^{3}$ , which in some sense measures the curvature of $M$ in space.

To construct the second fundamental form requires a small digression. After the digression we will discuss how it relates to the curvature of $M$ .

Construction of the second fundamental form

Consider the tangent planes $\mathrm{T}_{p}M$ of the surface $M$ for each point $p\in M$ . There are two unit normals to $\mathrm{T}_{p}M$ . Assuming $M$ is orientable, we can choose one of these unit normals (http://planetmath.org/MutualPositionsOfVectors), $n(p)$ , so that $n(p)$ varies smoothly with $p$ .

Since $n(p)$ is a unit vector in $\mathbb{R}^{3}$ , it may be considered as a point on the sphere $S^{2}\subset\mathbb{R}^{3}$ . Then we have a map $n\colon M\to S^{2}$ . It is called the normal map or Gauss map.

The second fundamental form is the tensor field $\mathcal{II}$ on $M$ defined by

$\mathcal{II}_{p}(\xi,\eta)=-\langle\operatorname{D}n_{p}(\xi),\eta\rangle\,,% \quad\xi,\eta\in\mathrm{T}_{p}M\,,$

(1)

where $\langle,\rangle$ is the dot product of $\mathbb{R}^{3}$ , and we consider the tangent planes of surfaces in $\mathbb{R}^{3}$ to be subspaces of $\mathbb{R}^{3}$ .

The linear transformation $\operatorname{D}n_{p}$ is in reality the tangent mapping $\operatorname{D}n_{p}\colon\mathrm{T}_{p}M\to\mathrm{T}_{n(p)}S^{2}$ , but since $\mathrm{T}_{n(p)}S^{2}=\mathrm{T}_{p}M$ by the definition of $n$ , we prefer to think of $\operatorname{D}n_{p}$ as $\operatorname{D}n_{p}\colon\mathrm{T}_{p}M\to\mathrm{T}_{p}M$ .

The tangent map $\operatorname{D}n$ , is often called the Weingarten map.

Proposition 1.

The second fundamental form is a symmetric form.

Proof.

This is a computation using a coordinate chart $\sigma$ for $M$ . Let $u, v$ be the corresponding names for the coordinates. From the equation

$\left\langle n,\frac{\partial\sigma}{\partial v}\right\rangle=0\,,$

differentiating with respect to $u$ using the product rule gives

$\begin{split}\displaystyle\left\langle n,\frac{\partial^{2}\sigma}{\partial u% \partial v}\right\rangle&\displaystyle=-\left\langle\frac{\partial n}{\partial u% },\frac{\partial\sigma}{\partial v}\right\rangle\\ &\displaystyle=-\left\langle\operatorname{D}n\left(\frac{\partial\sigma}{% \partial u}\right),\frac{\partial\sigma}{\partial v}\right\rangle\\ &\displaystyle=\mathcal{II}\left(\frac{\partial\sigma}{\partial u},\frac{% \partial\sigma}{\partial v}\right)\,.\end{split}$

(2)

(The second equality follows from the definition of the tangent map $\operatorname{D}n$ .) Reversing the roles of $u, v$ and repeating the last derivation, we obtain also:

$\left\langle n,\frac{\partial^{2}\sigma}{\partial u\partial v}\right\rangle=% \left\langle n,\frac{\partial^{2}\sigma}{\partial v\partial u}\right\rangle=% \mathcal{II}\left(\frac{\partial\sigma}{\partial v},\frac{\partial\sigma}{% \partial u}\right)\,.$

(3)

Since $\partial\sigma/\partial u$ and $\partial\sigma/\partial v$ form a basis for $\mathrm{T}_{p}M$ , combining (2) and (3) proves that $\mathcal{II}$ is symmetric. ∎

In view of Proposition 1, it is customary to regard the second fundamental form as a quadratic form, as it done with the first fundamental form. Thus, the second fundamental form is referred to with the following expression¹¹ Unfortunately the coefficient $M$ here clashes with our use of the letter $M$ for the surface (manifold), but whenever we write $M$ , the context should make clear which meaning is intended. The use of the symbols $L, M, N$ for the coefficients of the second fundamental form is standard, but probably was established long before anyone thought about manifolds.:

$L\,du^{2}+2M\,dudv+N\,dv^{2}\,.$

Compare with the tensor notation

$\mathcal{II}=L\,du\otimes du+M\,du\otimes dv+M\,dv\otimes du+Ndv\otimes dv\,.$

Or in matrix form (with respect to the coordinates $u, v$ ),

$\mathcal{II}=\begin{pmatrix}L&M\\ M&N\end{pmatrix}\,.$

Curvature of curves on a surface

Let $\gamma$ be a curve lying on the surface $M$ , parameterized by arc-length. Recall that the curvature $\kappa(s)$ of $\gamma$ at $s$ is $\gamma^{\prime\prime}(s)$ . If we want to measure the curvature of the surface, it is natural to consider the component of $\gamma^{\prime\prime}(s)$ in the normal $n(\gamma(s))$ . Precisely, this quantity is

$\langle\gamma^{\prime\prime}(s),n(\gamma(s))\rangle\,,$

and is called the normal curvature of $\gamma$ on $M$ .

So to study the curvature of $M$ , we ignore the component of the curvature of $\gamma$ in the tangent plane of $M$ . Also, physically speaking, the normal curvature is proportional to the acceleration required to keep a moving particle on the surface $M$ .

We now come to the motivation for defining the second fundamental form:

Proposition 2.

Let $\gamma$ be a curve on $M$ , parameterized by arc-length, and $\gamma(s)=p$ . Then

$\langle\gamma^{\prime\prime}(s),n(p)\rangle=\mathcal{II}\bigl{(}\gamma^{\prime% }(s),\gamma^{\prime}(s)\bigr{)}\,.$

Proof.

From the equation

$\langle n(\gamma(s)),\gamma^{\prime}(s)\rangle=0\,,$

differentiate with respect to $s$ :

	$\displaystyle\langle n(\gamma(s)),\gamma^{\prime\prime}(s)\rangle$	$\displaystyle=-\left\langle\frac{d}{ds}n(\gamma(s)),\gamma^{\prime}(s)\right\rangle$
		$\displaystyle=-\bigl{\langle}\operatorname{D}n(\gamma^{\prime}(s)),\gamma^{% \prime}(s)\bigr{\rangle}$
		$\displaystyle=\mathcal{II}\bigl{(}\gamma^{\prime}(s),\gamma^{\prime}(s)\bigr{)% }\,.\qed$

It is now time to mention an important consequence of Proposition 1: the fact that $\mathcal{II}$ is symmetric means that $-\operatorname{D}n$ is self-adjoint with respect to the inner product $\mathcal{I}$ (the first fundamental form). So, if $-\operatorname{D}n$ is expressed as a matrix with orthonormal coordinates (with respect to $\mathcal{I}$ ), then the matrix is symmetric. (The minus sign in front of $\operatorname{D}n$ is to make the formulas work out nicely.)

Certain theorems in linear algebra tell us that, $-\operatorname{D}n_{p}$ being self-adjoint, it has an orthonormal basis of eigenvectors $e_{1},e_{2}$ with corresponding eigenvalues $\kappa_{1}\leq\kappa_{2}$ . These eigenvalues are called the principal curvatures of $M$ at $p$ . The eigenvectors $e_{1},e_{2}$ are the principal directions. The terminology is justified by the following theorem:

Theorem 1 (Euler’s Theorem).

The normal curvature of a curve $\gamma$ has the form

$\langle\gamma^{\prime\prime}(s),n(p)\rangle=\kappa_{1}\,\cos^{2}\theta+\kappa_% {2}\,\sin^{2}\theta\,,\quad p=\gamma(s)\,.$

It follows that the minimum possible normal curvature is $\kappa_{1}$ , and the maximum possible is $\kappa_{2}$ .

Proof.

Since $e_{1},e_{2}$ form an orthonormal basis for $\mathrm{T}_{p}M$ , we may write

$\gamma^{\prime}(s)=\cos\theta\,e_{1}+\sin\theta\,e_{2}$

for some angle $\theta$ . Then

	$\displaystyle\langle\gamma^{\prime\prime}(s),n(p)\rangle$	$\displaystyle=\mathcal{II}\bigl{(}\gamma^{\prime}(s),\gamma^{\prime}(s)\bigr{)}$
		$\displaystyle=\bigl{\langle}-\operatorname{D}n_{p}(\gamma^{\prime}(s)),\gamma^% {\prime}(s)\bigr{\rangle}$
		$\displaystyle=\langle\kappa_{1}\cos\theta\,e_{1}+\kappa_{2}\sin\theta\,e_{2},% \cos\theta\,e_{1}+\sin\theta\,e_{2}\rangle$
		$\displaystyle=\kappa_{1}\,\cos^{2}\theta+\kappa_{2}\,\sin^{2}\theta\,.\qed$

Matrix representations of second fundamental form and Weingarten map

At this point, we should find the explicit prescriptions for calculating the second fundamental form and the Weingarten map.

Let $\sigma$ be a coordinate chart for $M$ , and $u, v$ be the names of the coordinates. For a test vector $\xi\in\mathrm{T}_{p}M$ , we write $\xi_{u}$ and $\xi_{v}$ for the $u, v$ coordinates of $\xi$ .

We compute the matrix $W$ for $-\operatorname{D}n$ in $u, v$ -coordinates. We have

	$\displaystyle\begin{pmatrix}\xi_{u}&\xi_{v}\end{pmatrix}\begin{pmatrix}L&M\\ M&N\end{pmatrix}\begin{pmatrix}\xi_{u}\\ \xi_{v}\end{pmatrix}$	$\displaystyle=\mathcal{II}(\xi,\xi)=\langle-\operatorname{D}n(\xi),\xi\rangle$
		$\displaystyle=\left(Q\begin{pmatrix}\xi_{u}\\ \xi_{v}\end{pmatrix}\right)^{\mathrm{T}}QW\begin{pmatrix}\xi_{u}\\ \xi_{v}\end{pmatrix}$
		$\displaystyle=\begin{pmatrix}\xi_{u}&\xi_{v}\end{pmatrix}\,(Q^{\mathrm{T}}Q)\,% W\begin{pmatrix}\xi_{u}\\ \xi_{v}\end{pmatrix}\,,$

where $Q$ is the matrix that changes from $u, v$ -coordinates to orthonormal coordinates for $\mathrm{T}_{p}M$ — this is necessary to compute the inner product. But

$Q^{\mathrm{T}}Q=\begin{pmatrix}E&F\\ F&G\end{pmatrix}=\mathcal{I}\quad\text{(the first fundamental form),}$

because $Q$ is the matrix with columns $\partial\sigma/\partial u$ and $\partial\sigma/\partial v$ expressed in orthonormal coordinates.

(More to be written…)

References

1 Michael Spivak. A Comprehensive Introduction to Differential Geometry, volumes I and II. Publish or Perish, 1979.
2 Andrew Pressley. Elementary Differential Geometry. Springer-Verlag, 2003.

Title	second fundamental form
Canonical name	SecondFundamentalForm
Date of creation	2013-03-22 15:29:02
Last modified on	2013-03-22 15:29:02
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	5
Author	stevecheng (10074)
Entry type	Definition
Classification	msc 53A05
Related topic	FirstFundamentalForm
Related topic	ShapeOperator
Related topic	NormalSection
Related topic	NormalCurvatures
Defines	normal curvature
Defines	principal direction
Defines	principal curvature
Defines	Weingarten matrix
Defines	Weingarten map
Defines	Gauss map