First fundamental form

It is the Riemannian metric defined on a local chart $(u,v)$ of a 2-dimensional Riemannian manifold (surface). It is usually written as

$$ d \widehat{s}^{2}=E d u^{2}+G d v^{2}+2 F d u d v $$

where $(u,v)$ are coordinates for the surface. The term $d \widehat{s}^{2}$ is nothing, but represents the squared length of a vector given in these coordinates by the pair $(du,dv)$. Indeed the first fundamental form, being a tensor field, should be written

$$ E du \otimes du+F du \otimes dv+F dv \otimes du+G dv \otimes dv $$

It has a visual interpretation in @needham2021visual page 35. If we write the first fundamental form as

$$ \mathrm{d} \widehat{\mathrm{s}}^{2}=\mathrm{A}^{2} \mathrm{du}^{2}+\mathrm{B}^{2} \mathrm{~d} v^{2}+2 \mathrm{~F} \mathrm{du} \mathrm{d} v, \text { where } \mathrm{F}=\mathrm{A} \mathrm{B} \cos \omega . $$

then

$A$ is the length expansion of a little displacement in the $u$ direction. That is, if in coordinates $(u,v)$ we move a quantity $\delta u$ along a curve $\{v=constant\}$, in the surface we will have moved a quantity $A\delta u$.

$B$ is the same in the $v$ direction.

$\omega$ is the angle sustained by the families of $u$-curves and $v$-curves.

Example

Sphere. First fundamental form in spherical coordinates is ds² = dφ² + sin²(φ) dθ².

Sphere. First fundamental form in Cartesian coordinates is ds² = (1 + x²/(1 - x² - u²)) dx² +2xu/(1 - x² - u²) dx du + (1 + u²/(1 - x² - u²)) du².

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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