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):

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

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