PlanetMath: first fundamental form

There are various notations for the first fundamental form; a common notation is $\mathcal{I}$ , for the roman letter one. Thus, \begin{equation}\label{simple} \mathcal{I}(v, w) = v \cdot w \end{equation}for vectors $v, w \in \real^3$ . (We consider the tangent planes of $M$ to be two-dimensional subspaces of $\real^3$ .)

Quadratic form representation

The letters $u$ and $v$ in (

) denote local coordinates on $M$ . Classically, $du$ and $dv$ meant ``infinitesimally small'' changes in $u$ and $v$ , but in modern differential geometry, $du$ and $dv$ have been given a precise meaning using differential forms.

In tensor notation, (

) is written as \begin{equation}\label{tensor} \mathcal{I} = E \, du \otimes du + F \, du \otimes dv + F \, dv \otimes du + G \, dv \otimes dv\,, \end{equation}Although the tensor notation is more clumsy, it allows us to rigorously justify a change of variables, by the rule $\alpha^* (du \otimes dv) = d(\alpha^*u) \otimes d(\alpha^*v)$ . See the example below.

The symbol $ds$ in (

) alludes to $$ ds^2 = dx^2 + dy^2 + dz^2\,, $$ the infinitesimal length of a curve. Compare with the modern notation \begin{equation}\label{cartesian} \mathcal{I} = dx \otimes dx + dy \otimes dy + dz \otimes dz\,. \end{equation}(This is just an alternate way of writing the definition of $\mathcal{I}$ : the restriction of the dot product on $\real^3$ .)

Example: sphere

$\displaystyle x$	$% latex2html id marker 410 $\displaystyle = \cos \phi \cos \theta$$
$\displaystyle y$	$% latex2html id marker 412 $\displaystyle = \cos \phi \sin \theta$$
$\displaystyle z$	$\displaystyle = \sin \phi\,,$

$% latex2html id marker 415 $\displaystyle \mathcal{I}$$	$% latex2html id marker 416 $\displaystyle = d (\cos \phi \cos \theta) \otimes d... ...s \phi \cos \theta) + d(\cos \phi \sin \theta) \otimes d(\cos \phi \sin \theta)$$
	$\displaystyle \quad + d(\sin \phi) \otimes d(\sin \phi)$
	$% latex2html id marker 418 $\displaystyle = (- \sin \phi \cos \theta \, d\phi -... ...) \otimes (- \sin \phi \cos \theta \, d\phi - \cos \phi \sin \theta \, d\theta)$$
	$% latex2html id marker 419 $\displaystyle \quad + (- \sin \phi \sin \theta \, d... ...) \otimes (- \sin \phi \sin \theta \, d\phi + \cos \phi \cos \theta \, d\theta)$$
	$\displaystyle \quad + (\cos \phi \, d\phi) \otimes (\cos \phi \, d\phi)$
	$% latex2html id marker 421 $\displaystyle = (\sin \phi \cos \theta)^2 \, d\phi \otimes d\phi + (\cos \phi \sin \theta)^2 \, d\theta \otimes d\theta$$
	$% latex2html id marker 422 $\displaystyle \quad + (\sin \phi \sin \theta)^2 \, d\phi \otimes d\phi + (\cos \phi \cos \theta)^2 \, d\theta \otimes d\theta$$
	$\displaystyle \quad + (\cos \phi)^2 \, d\phi \otimes d\phi$
(note that the cross terms with $% latex2html id marker 425 $ d\phi \otimes d\theta$$ and $% latex2html id marker 427 $ d\theta \otimes d\phi$$ cancel)
	$% latex2html id marker 428 $\displaystyle = \sin^2 \phi \, d\phi \otimes d\phi + \cos^2 \phi \, d\theta \otimes d\theta + \cos^2 \phi \, d\phi \otimes d\phi$$
	$% latex2html id marker 429 $\displaystyle = d\phi \otimes d\phi + \cos^2 \phi \, d\theta \otimes d\theta\,.$$

Use of first fundamental form to compute lengths and areas

The first fundamental form is related to the area form. If $$ ds^2 = E \, du^2 + 2F \, du dv + G \, dv^2 $$ then $$ dA = \sqrt{EG-F^2} \: du \wedge dv $$ is the area form. For the sphere, this is $$ dA = \sqrt{\cos^2 \phi - 0} \: d\phi \wedge d\theta = \cos \phi \: d\phi \wedge d\theta\,, $$ which is just the formula given in calculus for evaluating surface integrals on the sphere using spherical coordinates.

The first fundamental form itself may be used to find the length $s$ of a curve $\gamma$ on a surface $M$ , when $\gamma$ is parameterized by local coordinates:

$\displaystyle s$	$\displaystyle = \int_\gamma ds = \int_\gamma \sqrt{ds^2}$
	$\displaystyle = \int_\gamma \sqrt{E \, du^2 + 2F \, du dv + G \, dv^2} \,, \quad \gamma(t) = (u,v)\,,$
	$\displaystyle = \int_a^b \sqrt{E \, \left(\frac{du}{dt}\right)^2 + 2F \, \frac{du}{dt} \frac{dv}{dt} + G \, \left(\frac{dv}{dt} \right)^2} \, dt\,.$

Example: plane and cylinder

$\displaystyle x$	$\displaystyle = \cos u$
$\displaystyle y$	$\displaystyle = \sin u$
$\displaystyle z$	$\displaystyle = v$

Relation with isometric maps

Of course, (

) and (

) are expressions of the first fundamental form in local coordinates of two different surfaces, so it makes no sense to say they are equal. But it is not hard to see that:

Suppose $\Phi\colon M \to N$ is an isometry of two surfaces, and $u, v$ are coordinates on $M$ . If we use the coordinates $u' = u \circ \Phi^{-1}$ and $v' = v \circ \Phi^{-1}$ on $N$ , then the first fundamental form of $N$ is obtained by taking the first fundamental form of $M$ and renaming $u, v$ to $u', v'$ .

Relation with conformal, equiareal maps

There is also a notion of a conformal mapping: a diffeomorphism $\Phi \colon M \to N$ is called conformal if $\Phi$ preserves the first fundamental form up to a non-zero constant of proportionality. (The proportion may vary at each point of $M$ and $N$ .) It may be verified that this is the same as saying that $\Phi$ preserves the angles of intersecting tangent vectors.

Yet another notion is that of an equiareal mapping: a diffeomorphism $\Phi \colon M \to N$ is called equiareal if $\Phi$ preserves preserves areas of all subregions of the surfaces. This amounts to saying that the quantity $\sqrt{EG - F^2}$ is invariant under $\Phi$ (provided we rename the variables as explained above).

For example, the projection of the sphere to the cylinder wrapping it is equiareal. This fact was used by Archimedes to show the sphere of radius $r$ has area $4\pi r^2$ , because the cylinder that wraps it also has area $4\pi r^2$ .

If $\Phi$ is both conformal and equiareal, then it is an isometry. As a well-known example, a sphere is not isometric to the plane, not even locally, so we cannot draw maps of the Earth that preserve both directions and relative proportion of lands. We must give up at least one of these properties: e.g. the Mercator projection preserves direction only; maps with Mercator look ``strange'' the first time one sees them, because such maps do not preserve area.

Relation with Gaussian curvature

This formula is known as Brioschi's formula; Brioschi had stated it without proof in 1854, and later it was calculated by Gauss.

This theorem is not obvious, since the usual definitions of the Gaussian curvature are not invariant (they depend on the particular embedding of the surface in $\real^3$ ).