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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. 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 ξa of the submanifold while the functions Xμ(ξa) encode the embedding into the higher-dimensional manifold whose tangent indices are denoted μ,ν.

See also: first fundamental form.

01-04-2007 01:16:19
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