Induced metric
Template:Short description In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback.<ref>Template:Cite book</ref> It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation:<ref>Template:Cite book</ref>
- <math>g_{ab} = \partial_a X^\mu \partial_b X^\nu g_{\mu\nu}\ </math>
Here <math>a</math>, <math>b</math> describe the indices of coordinates <math>\xi^a</math> of the submanifold while the functions <math>X^\mu(\xi^a)</math> encode the embedding into the higher-dimensional manifold whose tangent indices are denoted <math>\mu</math>, <math>\nu</math>.
Example – Curve in 3DEdit
Let
- <math>
\Pi\colon \mathcal{C} \to \mathbb{R}^3,\ \tau \mapsto \begin{cases}\begin{align}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{align} \end{cases}</math>
be a map from the domain of the curve <math>\mathcal{C}</math> with parameter <math>\tau</math> into the Euclidean manifold <math>\mathbb{R}^3</math>. Here <math>a,b,m,n\in\mathbb{R}</math> are constants.
Then there is a metric given on <math>\mathbb{R}^3</math> as
- <math>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} </math>.
and we compute
- <math>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
</math>
Therefore <math>g_\mathcal{C}=(m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2) \, \mathrm{d}\tau\otimes \mathrm{d}\tau</math>