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, through pullback inducing. It may be calculated using the following formula, which is the component form of the pullback operation:
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.Let
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
Therefore
