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. It may be calculated using the following formula (written using Einstein summation convention):

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

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