词条 | Induced metric |
释义 |
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 torusLet 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
1 : Differential geometry |
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