Induced metric

In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold which is calculated from the metric tensor on a larger manifold into which the submanifold is embedded, through pullback inducing. It may be calculated using the following formula (written using Einstein summation convention), which is the component form of the pullback operation:[1]

${\displaystyle g_{ab}=\partial _{a}X^{\mu }\partial _{b}X^{\nu }g_{\mu \nu }\ }$

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

Example - Curve on a torus

Let

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

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

Then there is a metric given on ${\displaystyle \mathbb {R} ^{3}}$ as

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

and we compute

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

Therefore ${\displaystyle g_{\mathcal {C}}=(m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2})\mathrm {d} \tau \otimes \mathrm {d} \tau }$

See also

• First fundamental form

References

1. Poisson, Eric (2004). A Relativist's Toolkit. Cambridge University Press. p. 62. ISBN 978-0-521-83091-1.