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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