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]
\( 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 on a torus
Let
\( \Pi \colon {\mathcal {C}}\to \mathbb {R} ^{3},\ \tau \mapsto {\begin{cases}{\begin{aligned}x^{1}&=(a+b\cos(n\cdot \tau ))\cos(m\cdot \tau )\\x^{2}&=(a+b\cos(n\cdot \tau ))\sin(m\cdot \tau )\\x^{3}&=b\sin(n\cdot \tau ).\end{aligned}}\end{cases}} \)
be a map from the domain of the curve \( {\mathcal {C}} \) 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
\( g_{\tau \tau }=\sum \limits _{\mu ,\nu }{\frac {\partial x^{\mu }}{\partial \tau }}{\frac {\partial x^{\nu }}{\partial \tau }}\underbrace {g_{\mu \nu }} _{\delta _{\mu \nu }}=\sum \limits _{\mu }\left({\frac {\partial x^{\mu }}{\partial \tau }}\right)^{2}=m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2} \)
Therefore \( g_{\mathcal {C}}=(m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2})\mathrm {d} \tau \otimes \mathrm {d} \tau \)
See also
First fundamental form
References
Poisson, Eric (2004). A Relativist's Toolkit. Cambridge University Press. p. 62. ISBN 978-0-521-83091-1.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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