Induced metric

In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback.[1] It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation:[2]

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 in 3D

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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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References

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  1. ^ Lee, John M. (2006-04-06). Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics. Springer Science & Business Media. pp. 25–27. ISBN 978-0-387-22726-9. OCLC 704424444.
  2. ^ Poisson, Eric (2004). A Relativist's Toolkit. Cambridge University Press. p. 62. ISBN 978-0-521-83091-1.