Induced metric

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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[edit]

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

See also[edit]