Induced metric

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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]

g a b = ∂ a X μ ∂ b X ν g μ ν   {\displaystyle g_{ab}=\partial _{a}X^{\mu }\partial _{b}X^{\nu }g_{\mu \nu }\ } {\displaystyle g_{ab}=\partial _{a}X^{\mu }\partial _{b}X^{\nu }g_{\mu \nu }\ }

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

Example – Curve in 3D

Let

Π : C → R 3 ,   τ ↦ { x 1 = ( a + b cos ⁡ ( n ⋅ τ ) ) cos ⁡ ( m ⋅ τ ) x 2 = ( a + b cos ⁡ ( n ⋅ τ ) ) sin ⁡ ( m ⋅ τ ) x 3 = b sin ⁡ ( n ⋅ τ ) . {\displaystyle \Pi \colon {\mathcal {C}}\to \mathbb {R} ^{3},\ \tau \mapsto {\begin{cases}{\begin{aligned}x^{1}&=(a+b\cos(n\cdot \tau ))\cos(m\cdot \tau )\\x^{2}&=(a+b\cos(n\cdot \tau ))\sin(m\cdot \tau )\\x^{3}&=b\sin(n\cdot \tau ).\end{aligned}}\end{cases}}} {\displaystyle \Pi \colon {\mathcal {C}}\to \mathbb {R} ^{3},\ \tau \mapsto {\begin{cases}{\begin{aligned}x^{1}&=(a+b\cos(n\cdot \tau ))\cos(m\cdot \tau )\\x^{2}&=(a+b\cos(n\cdot \tau ))\sin(m\cdot \tau )\\x^{3}&=b\sin(n\cdot \tau ).\end{aligned}}\end{cases}}}

be a map from the domain of the curve C {\displaystyle {\mathcal {C}}} {\displaystyle {\mathcal {C}}} with parameter τ {\displaystyle \tau } {\displaystyle \tau } into the Euclidean manifold R 3 {\displaystyle \mathbb {R} ^{3}} {\displaystyle \mathbb {R} ^{3}}. Here a , b , m , n ∈ R {\displaystyle a,b,m,n\in \mathbb {R} } {\displaystyle a,b,m,n\in \mathbb {R} } are constants.

Then there is a metric given on R 3 {\displaystyle \mathbb {R} ^{3}} {\displaystyle \mathbb {R} ^{3}} as

g = ∑ μ , ν g μ ν d x μ ⊗ d x ν with g μ ν = ( 1 0 0 0 1 0 0 0 1 ) {\displaystyle g=\sum \limits _{\mu ,\nu }g_{\mu \nu }\mathrm {d} x^{\mu }\otimes \mathrm {d} x^{\nu }\quad {\text{with}}\quad g_{\mu \nu }={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}} {\displaystyle g=\sum \limits _{\mu ,\nu }g_{\mu \nu }\mathrm {d} x^{\mu }\otimes \mathrm {d} x^{\nu }\quad {\text{with}}\quad g_{\mu \nu }={\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}.

and we compute

g τ τ = ∑ μ , ν ∂ x μ ∂ τ ∂ x ν ∂ τ g μ ν ⏟ δ μ ν = ∑ μ ( ∂ x μ ∂ τ ) 2 = m 2 a 2 + 2 m 2 a b cos ⁡ ( n ⋅ τ ) + m 2 b 2 cos 2 ⁡ ( n ⋅ τ ) + b 2 n 2 {\displaystyle g_{\tau \tau }=\sum \limits _{\mu ,\nu }{\frac {\partial x^{\mu }}{\partial \tau }}{\frac {\partial x^{\nu }}{\partial \tau }}\underbrace {g_{\mu \nu }} _{\delta _{\mu \nu }}=\sum \limits _{\mu }\left({\frac {\partial x^{\mu }}{\partial \tau }}\right)^{2}=m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2}} {\displaystyle g_{\tau \tau }=\sum \limits _{\mu ,\nu }{\frac {\partial x^{\mu }}{\partial \tau }}{\frac {\partial x^{\nu }}{\partial \tau }}\underbrace {g_{\mu \nu }} _{\delta _{\mu \nu }}=\sum \limits _{\mu }\left({\frac {\partial x^{\mu }}{\partial \tau }}\right)^{2}=m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2}}

Therefore g C = ( m 2 a 2 + 2 m 2 a b cos ⁡ ( n ⋅ τ ) + m 2 b 2 cos 2 ⁡ ( n ⋅ τ ) + b 2 n 2 ) d τ ⊗ d τ {\displaystyle g_{\mathcal {C}}=(m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2})\,\mathrm {d} \tau \otimes \mathrm {d} \tau } {\displaystyle g_{\mathcal {C}}=(m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2})\,\mathrm {d} \tau \otimes \mathrm {d} \tau }

See also

References

  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.