# Ngô Quốc Anh

## December 12, 2009

### R-G: The First Fundamental Form (Riemannian Metric)

Filed under: Riemannian geometry — Ngô Quốc Anh @ 16:42

Given a curve $C$ on a surface $X$ parametrized by two parameters $u, v$, we first compute the element of arc length of the curve $C$. For this, we need to compute the square norm of the tangent vector $\dot C (t)$. The square norm of the tangent vector $\dot C (t)$ to the curve $C$ at $p$ is $\displaystyle \| \dot C \|^2 = (X_u \dot u + X_v \dot v) \cdot (X_u \dot u + X_v \dot v)$,

where $\cdot$ is the inner product in $\mathbb R^3$, and thus, $\displaystyle \| \dot C \|^2 = (X_u\cdot X_u)\dot u^2 + 2(X_u\cdot X_v) \dot u\dot v +(X_v \cdot X_v) \dot v^2$.

Following common usage, we let $\displaystyle E = X_u \cdot X_u, \quad F = X_u \cdot X_v, \quad G = X_v \cdot X_v$,

and $\displaystyle \| \dot C \|^2 = E \dot u^ 2 + 2F\dot u \dot v+ G\dot v^2$.

Euler already obtained this formula in 1760. Thus, the map $\displaystyle (x, y) \mapsto Ex^2 + 2Fxy + Gy^2$

is a quadratic form on $\mathbb R^2$, and since it is equal to $\| \dot C \|^2$, it is positive definite. This quadratric form plays a major role in the theory of surfaces, and deserves an official definition.

Definition. Given a surface $X$, for any point $p =X(u, v)$ on $X$, letting $\displaystyle E = X_u \cdot X_u, \quad F = X_u \cdot X_v, \quad G = X_v \cdot X_v$,

the positive definite quadratic form $\displaystyle (x, y) \mapsto Ex^2 + 2Fxy + Gy^2$

is called the first fundamental form of $X$ at $p$. It is often denoted as $\mathrm I_p$, and in matrix form, we have $\displaystyle {I_p}\left( {x,y} \right) = \left( {\begin{array}{*{20}{c}} x & y\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} E & F\\ F & G\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} x\\ y\\ \end{array} } \right)$.

The symmetric bilinear form $\varphi_I$ associated with $I$ is an inner product on the tangent space at $p$, such that $\displaystyle {\varphi_I}\left( ({x_1,y_1}),(x_2,y_2) \right) = \left( {\begin{array}{*{20}{c}} x_1 & y_1\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} E & F\\ F & G\\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} x_2\\ y_2\\ \end{array} } \right)$.

This inner product is also denoted as ${\left\langle {\left( {{x_1},{y_1}} \right),{\text{ }}\left( {{x_2},{y_2}} \right)} \right\rangle _p}$. The inner product $\varphi_I$ can be used to determine the angle of two curves passing through $p$, i.e., the angle $\theta$ of the tangent vectors to these two curves at $p$. We have $\displaystyle \cos \theta = \frac{{\left\langle {\left( {{\dot u_1},{\dot v_1}} \right),{\text{ }}\left( {{\dot u_2},{\dot v_2}} \right)} \right\rangle }}{{\sqrt {I\left( {{\dot u_1},{\dot v_1}} \right)} \sqrt {I\left( {{\dot u_2},{\dot v_2}} \right)} }}$.

For example, the angle between the $u$-curve and the $v$-curve passing through $p$ (where $u$ or $v$ is constant) is given by $\displaystyle\cos \theta= \frac{F}{{\sqrt {EG} }}$.

Thus, the $u$-curves and the $v$-curves are orthogonal iff $F(u, v) =0$ on $\Omega$.

Example 1. The helicoid is the surface defined over $\mathbb R \times \mathbb R$ such that $\displaystyle x=u \cos v, y=u \sin v, z=v$.

This is the surface generated by a line parallel to the $xOy$ plane, touching the $z$ axis, and also touching an helix of axis $Oz$. It is easily verified that $(E, F,G) = (1, 0, u^2+ 1)$. The figure below shows a portion of helicoid corresponding to $0 \leq v \leq 2\pi$ and $-2\leq u \leq 2$. Example 2. The catenoid is the surface of revolution defined over $\mathbb R \times \mathbb R$ such that $\displaystyle x=\cosh u \cos v, y=\cosh u \sin v, z=u$.

It is the surface obtained by rotating a catenary around the $z$-axis. It is easily verified that $(E, F,G) = (\cosh^2 u, 0, \cosh^2 u)$. The figure below shows a portion of catenoid corresponding to $0 \leq v \leq 2\pi$ and $-2\leq u \leq 2$. We will see how the first fundamental form relates to the curvature of curves on a surface.

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