Induced Metric
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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 (written using Einstein summation convention), 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 .

Example - Curve on a torus

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


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