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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):

$g_{ab}={\partial }_a{X}^{\mu }{\partial }_b{X}^{\nu }{g}_{\mu \nu }(X^{\alpha })$

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

$\begin{array}{rl}\mathrm{\Pi }:\mathcal{𝒞}& \to {\mathbb{ℝ}}^3\\ \tau & \mapsto \{\begin{array}{c}{x}^1=(a+b\cos (n\cdot \tau ))\cos (m\cdot \tau )\\ x^2=(a+b\cos (n\cdot \tau ))\sin (m\cdot \tau )\\ x^3=b\sin (n\cdot \tau )\end{array}\end{array}$

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

$g=\sum _{\mu ,\nu }{g}_{\mu \nu }\mathrm{d}{x}^{\mu }\otimes \mathrm{d}{x}^{\nu }\text{with}{g}_{\mu \nu }=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$.

and we compute

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

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

See also

• First fundamental form

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