induced metric

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.{{Cite book|last=Lee|first=John M.|url=https://books.google.com/books?id=92PgBwAAQBAJ|title=Riemannian Manifolds: An Introduction to Curvature|date=2006-04-06|publisher=Springer Science & Business Media|isbn=978-0-387-22726-9|series=Graduate Texts in Mathematics|pages=25–27|language=en|oclc=704424444}} It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation:{{cite book |last1=Poisson |first1=Eric |date=2004 |title=A Relativist's Toolkit |publisher=Cambridge University Press |page=62 |isbn=978-0-521-83091-1 }}

: $g_{ab} = \partial_a X^\mu \partial_b X^\nu g_{\mu\nu}\$

Here $a$ , $b$ describe the indices of coordinates $\xi^a$ of the submanifold while the functions $X^\mu(\xi^a)$ encode the embedding into the higher-dimensional manifold whose tangent indices are denoted $\mu$ , $\nu$ .

Example – Curve in 3D

Let

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

be a map from the domain of the curve $\mathcal{C}$ with parameter $\tau$ into the Euclidean manifold $\mathbb{R}^3$ . Here $a,b,m,n\in\mathbb{R}$ are constants.

Then there is a metric given on $\mathbb{R}^3$ 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 $g_\mathcal{C}=(m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2) \, \mathrm{d}\tau\otimes \mathrm{d}\tau$

References

Category:Differential geometry

induced metric

Example – Curve in 3D

See also

References