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

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