Induced Metric - Example - Curve On A Torus

Curve On A Torus

Let

$\begin{align} \Pi\colon \mathcal{C} &\to \mathbb{R}^3 \\ \tau &\mapsto \left\{\quad\begin{matrix}x^1=(a+b\cos(n\cdot \tau))\cos(m\cdot \tau)\\x^2=(a+b\cos(n\cdot \tau))\sin(m\cdot \tau)\\x^3=b\sin(n\cdot \tau)\end{matrix}\right. \end{align}$

be a map from the domain of the curve with parameter into the euclidean manifold . Here are constants.

Then there is a metric given on as

$g=\sum\limits_{\mu,\nu}g_{\mu\nu}\mathrm{d}x^\mu\otimes \mathrm{d}x^\nu\quad\text{with}\quad g_{\mu\nu} = \begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end{pmatrix}$ .

and we compute

$g_{\tau\tau}=\sum\limits_{\mu,\nu}\frac{\partial x^\mu}{\partial \tau}\frac{\partial x^\nu}{\partial \tau}\underbrace{g_{\mu\nu}}_{\delta_{\mu\nu}} = \sum\limits_\mu\left(\frac{\partial x^\mu}{\partial \tau}\right)^2=m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2$

Therefore

Read more about this topic: Induced Metric, Example

Famous quotes containing the word curve:

“Nothing ever prepares a couple for having a baby, especially the first one. And even baby number two or three, the surprises and challenges, the cosmic curve balls, keep on coming. We can’t believe how much children change everything—the time we rise and the time we go to bed; the way we fight and the way we get along. Even when, and if, we make love.”
—Susan Lapinski (20th century)