Induced metric

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 ${\displaystyle \mu }$, ${\displaystyle \nu }$.