Definitions

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:

$g_\left\{ab\right\} = partial_a X^mu partial_b X^nu g_\left\{munu\right\} \left(X^alpha\right)$

Here $a,b$ describe the indices of coordinates $xi^a$ of the submanifold while the functions $X^mu\left(xi^a\right)$ encode the embedding into the higher-dimensional manifold whose tangent indices are denoted $mu,nu$.