induced metric について

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induced metric ：ウィキペディア英語版

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 (written using Einstein summation convention):
:

g_ = \partial_a X^\mu \partial_b X^\nu  g_\

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 \

.
==Example - Curve on a torus==

Let
:

\Pi\colon \mathcal \to \mathbb^3,\ \tau \mapsto \begin\beginx^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 \end

be a map from the domain of the curve

\mathcal

with parameter

\tau

into the euclidean manifold

\mathbb^3

. Here

a,b,m,n\in\mathbb

are constants.
Then there is a metric given on

\mathbb^3

as
:

g=\sum\limits_g_\mathrmx^\mu\otimes \mathrmx^\nu\quad\text\quadg_ = \begin1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 1\end

.
and we compute
:

g_=\sum\limits_\frac\frac\underbrace} = \sum\limits_\mu\left(\frac\right)^2=m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2

Therefore

g_\mathcal=(m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2)\mathrm\tau\otimes \mathrm\tau

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