PlanetMath: first fundamental form

There are various notations for the first fundamental form; a common notation is $% latex2html id marker 687 $ \mathcal{I}$$ , for the roman letter one. Thus,

$% latex2html id marker 689 $\displaystyle \mathcal{I}(v, w) = v \cdot w$$

(1)

Quadratic form representation

$\displaystyle T(u, v) = \frac{1}{2} \bigl( Q(u+v) - Q(u) - Q(v) \bigr)\,, \quad Q(w) = T(w,w)\,,$

$\displaystyle ds^2 = E \, du^2 + 2F \, du dv + G \, dv^2\,.$

(2)

$% latex2html id marker 742 $\displaystyle \mathcal{I} = E \, du \otimes du + F \, du \otimes dv + F \, dv \otimes du + G \, dv \otimes dv\,,$$

(3)

$\displaystyle ds^2 = dx^2 + dy^2 + dz^2\,,$

$% latex2html id marker 750 $\displaystyle \mathcal{I} = dx \otimes dx + dy \otimes dy + dz \otimes dz\,.$$

(4)

Example: sphere

$\displaystyle x$	$% latex2html id marker 761 $\displaystyle = \cos \phi \cos \theta$$
$\displaystyle y$	$% latex2html id marker 763 $\displaystyle = \cos \phi \sin \theta$$
$\displaystyle z$	$\displaystyle = \sin \phi\,,$

$% latex2html id marker 766 $\displaystyle \mathcal{I}$$	$% latex2html id marker 767 $\displaystyle = d (\cos \phi \cos \theta) \otimes d... ...s \phi \cos \theta) + d(\cos \phi \sin \theta) \otimes d(\cos \phi \sin \theta)$$
	$\displaystyle \quad + d(\sin \phi) \otimes d(\sin \phi)$
	$% latex2html id marker 769 $\displaystyle = (- \sin \phi \cos \theta \, d\phi -... ...) \otimes (- \sin \phi \cos \theta \, d\phi - \cos \phi \sin \theta \, d\theta)$$
	$% latex2html id marker 770 $\displaystyle \quad + (- \sin \phi \sin \theta \, d... ...) \otimes (- \sin \phi \sin \theta \, d\phi + \cos \phi \cos \theta \, d\theta)$$
	$\displaystyle \quad + (\cos \phi \, d\phi) \otimes (\cos \phi \, d\phi)$
	$% latex2html id marker 772 $\displaystyle = (\sin \phi \cos \theta)^2 \, d\phi \otimes d\phi + (\cos \phi \sin \theta)^2 \, d\theta \otimes d\theta$$
	$% latex2html id marker 773 $\displaystyle \quad + (\sin \phi \sin \theta)^2 \, d\phi \otimes d\phi + (\cos \phi \cos \theta)^2 \, d\theta \otimes d\theta$$
	$\displaystyle \quad + (\cos \phi)^2 \, d\phi \otimes d\phi$
(note that the cross terms with $% latex2html id marker 776 $ d\phi \otimes d\theta$$ and $% latex2html id marker 778 $ d\theta \otimes d\phi$$ cancel)
	$% latex2html id marker 779 $\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$$
	$% latex2html id marker 780 $\displaystyle = d\phi \otimes d\phi + \cos^2 \phi \, d\theta \otimes d\theta\,.$$

$% latex2html id marker 790 $\displaystyle d\phi^2 + \cos^2 \phi \, d\theta^2 $$

Use of first fundamental form to compute lengths and areas

$\displaystyle ds^2 = E \, du^2 + 2F \, du dv + G \, dv^2$

$\displaystyle dA = \sqrt{EG-F^2} \: du \wedge dv$

$% latex2html id marker 797 $\displaystyle dA = \sqrt{\cos^2 \phi - 0} \: d\phi \wedge d\theta = \cos \phi \: d\phi \wedge d\theta\,, $$

The first fundamental form itself may be used to find the length

of a curve $\gamma$ on a surface

, 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

$\displaystyle ds^2 = dx^2 + dy^2\,.$

(5)

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

$\displaystyle ds^2 = du^2 + dv^2\,.$

(6)

Relation with isometric maps

Of course, (5) and (6) 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

are coordinates on

. If we use the coordinates $u' = u \circ \Phi^{-1}$ and $v' = v \circ \Phi^{-1}$ on

, then the first fundamental form of

is obtained by taking the first fundamental form of

and renaming

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

and

.) 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

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

$\displaystyle K = \frac{\left\lvert \begin{matrix} -\frac{1}{2} E_{vv} + F_{uv}... ... E_v & E & F \ \frac{1}{2} G_u & F & G \end{matrix}\right\rvert}{(EG-F^2)^2}$

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 $% latex2html id marker 887 $ \mathbb{R}^3$$ ).