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:

:$g\_\{ab\}\; =\; partial\_a\; X^mu\; partial\_b\; X^\; u\; g\_\{mu\; u\}\; (X^alpha)$

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 $mu,\; u$.

ee also

*First fundamental form.