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 describe the indices of coordinates of the submanifold while the functions encode the embedding into the higher-dimensional manifold whose tangent indices are denoted .
Example - Curve on a torus
be a map from the domain of the curve with parameter into the Euclidean manifold . Here are constants.
Then there is a metric given on as
and we compute