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:

g_{ab} = \partial_a X^\mu \partial_b X^\nu g_{\mu\nu} (X^\alpha) \

Here a,b \ describe the indices of coordinates \xi^a \ of the submanifold while the functions X^\mu(\xi^a) \ encode the embedding into the higher-dimensional manifold whose tangent indices are denoted \mu,\nu \ .

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