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
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
References
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