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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. It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation:

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

Here ${\displaystyle a}$, ${\displaystyle b}$ 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 ${\displaystyle \mu }$, ${\displaystyle \nu }$.

Example – Curve in 3D

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. Lee, John M. (2006-04-06). Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics. Springer Science & Business Media. pp. 25–27. ISBN 978-0-387-22726-9. OCLC 704424444.
2. Poisson, Eric (2004). A Relativist's Toolkit. Cambridge University Press. p. 62. ISBN 978-0-521-83091-1.

• Differential geometry