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.^[1] It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation:^[2]

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

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