PlanetMath: second fundamental form

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

Construction of the second fundamental form

Since

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 linear transformation ${\mathrm{D}}n_p$ is in reality the tangent mapping ${\mathrm{D}}n_p \colon \mathrm{T}_p M\to \mathrm{T}_{ n(p)} S^2$ , but since $\mathrm{T}_{ n(p)}S^2 = \mathrm{T}_p M$ by the definition of

, we prefer to think of ${\mathrm{D}}n_p$ as ${\mathrm{D}}n_p \colon \mathrm{T}_p M\to \mathrm{T}_p M$ .

Proposition 1 The second fundamental form is a symmetric form.

Proof. This is a computation using a coordinate chart $\sigma$ for

. Let

be the corresponding names for the coordinates. From the equation

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

differentiating with respect to

using the product rule gives

$\begin{displaymath}\begin{split}\left\langle n, \frac{\partial^2 \sigma}{\partia... ...al u}, \frac{\partial \sigma}{\partial v}\right)\,. \end{split}\end{displaymath}$

(2)

(The second equality follows from the definition of the tangent map ${\mathrm{D}}n$ .) Reversing the roles of

and repeating the last derivation, we obtain also:

$\displaystyle \left\langle n, \frac{\partial^2 \sigma}{\partial u \partial v} \... ...frac{\partial \sigma}{\partial v}, \frac{\partial \sigma}{\partial u}\right)\,.$

(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. $\qedsymbol$

Curvature of curves on a surface

So to study the curvature of

, we ignore the component of the curvature of $\gamma$ in the tangent plane of

. Also, physically speaking, the normal curvature is proportional to the acceleration required to keep a moving particle on the surface

Proposition 2 Let $\gamma$ be a curve on , parameterized by arc-length, and $\gamma(s) = p$ . Then

$\displaystyle \langle \gamma''(s), n(p) \rangle = \mathcal{II}\bigl(\gamma'(s), \gamma'(s)\bigr)\,.$

Proof. From the equation

$\displaystyle \langle n(\gamma(s)), \gamma'(s) \rangle = 0\,,$

differentiate with respect to

$\displaystyle \langle n(\gamma(s)), \gamma''(s) \rangle$	$\displaystyle = - \left\langle \frac{d}{ds} n(\gamma(s)), \gamma'(s) \right\rangle$
	$\displaystyle = -\bigl\langle {\mathrm{D}}n(\gamma'(s)), \gamma'(s) \bigr\rangle$
	$\displaystyle = \mathcal{II}\bigl(\gamma'(s), \gamma'(s)\bigr)\,. \qedhere$

$\qedsymbol$

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

Certain theorems in linear algebra tell us that, $-{\mathrm{D}}n_p$ being self-adjoint, it has an orthonormal basis of eigenvectors

with corresponding eigenvalues $\kappa_1 \leq \kappa_2$ . These eigenvalues are called the principal curvatures of

. The eigenvectors

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

$\displaystyle \langle \gamma''(s), n(p) \rangle = \kappa_1 \, \cos^2 \theta + \kappa_2 \, \sin^2 \theta\,, \quad p = \gamma(s)\,.$

It follows that the minimum possible normal curvature is $\kappa_1$ , and the maximum possible is $\kappa_2$ .

Proof. Since

form an orthonormal basis for $\mathrm{T}_p M$ , we may write

$\displaystyle \gamma'(s) = \cos \theta \, e_1 + \sin \theta \, e_2$

for some angle $\theta$ . Then

$\displaystyle \langle \gamma''(s), n(p) \rangle$	$\displaystyle = \mathcal{II}\bigl(\gamma'(s), \gamma'(s) \bigr)$
	$\displaystyle = \bigl\langle -{\mathrm{D}}n_p(\gamma'(s)), \gamma'(s) \bigr\rangle$
	$\displaystyle = \langle \kappa_1 \cos \theta \, e_1 + \kappa_2 \sin \theta \, e_2, \cos \theta \, e_1 + \sin \theta \, e_2 \rangle$
	$\displaystyle = \kappa_1 \, \cos^2 \theta + \kappa_2 \, \sin^2 \theta\,. \qedhere$