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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_{ab}=\partial _{a}X^{\mu }\partial _{b}X^{\nu }g_{\mu \nu }(X^{\alpha })\

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

{\begin{aligned}\Pi \colon {\mathcal {C}}&\to \mathbb {R} ^{3}\\\tau &\mapsto \left\{\quad {\begin{matrix}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{matrix}}\right.\end{aligned}}

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

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

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_{\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 $g_{\mathcal {C}}=(m^{2}a^{2}+2m^{2}ab\cos(n\cdot \tau )+m^{2}b^{2}\cos ^{2}(n\cdot \tau )+b^{2}n^{2})\mathrm {d} \tau \otimes \mathrm {d} \tau$

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it:Metrica indotta ru:Индуцированная метрика

@@ Line 1: / Line 1: @@
-[[Image:Centered decagonal number.svg|280px|right]]
+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]]):
-A '''centered decagonal number''' is a [[centered number|centered]] [[figurate number]] that represents a [[decagon]] with a dot in the center and all other dots surrounding the center dot in successive decagonal layers. The centered decagonal number for ''n'' is given by the formula
+:<math>g_{ab} = \partial_a X^\mu \partial_b X^\nu  g_{\mu\nu} (X^\alpha) \ </math>
-: <math>5(n^2-n)+1 \, </math>
+Here <math>a,b \ </math> describe the indices of coordinates <math>\xi^a \ </math> of the submanifold while the functions <math>X^\mu(\xi^a) \ </math> encode the embedding into the higher-dimensional manifold whose tangent indices are denoted <math>\mu,\nu \ </math>.
-Thus, the first few centered decagonal numbers are
+==Example - Curve on a torus==
-: [[1 (number)|1]], [[11 (number)|11]], [[31 (number)|31]], [[61 (number)|61]], [[101 (number)|101]], [[151 (number)|151]], 211, 281, 361, 451, 551, 661, 781, [[911 (number)|911]], 1051, ... {{OEIS|id=A062786}}
+Let
+: <math>\begin{align}
+ \Pi\colon \mathcal{C} &\to \mathbb{R}^3 \\
+ \tau &\mapsto \left\{\quad\begin{matrix}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{matrix}\right.
+\end{align}</math>
-Like any other centered ''k''-gonal number, the ''n''th centered decagonal number can reckoned by multiplying the (''n''&nbsp;&minus;&nbsp;1)th [[triangular number]] by ''k'', 10 in this case, then adding 1. As a consequence of performing the calculation in base 10, the centered decagonal numbers can be obtained by simply adding a 1 to the right of each triangular number. Therefore, all centered decagonal numbers are odd and in base 10 always end in&nbsp;1.
+be a map from the domain of the curve <math>\mathcal{C}</math>  with parameter <math>\tau</math> into the euclidean manifold <math>\mathbb{R}^3</math>. Here <math>a,b,m,n\in\mathbb{R}</math> are constants.
-Another consequence of this relation to triangular numbers is the simple [[recurrence relation]] for centered decagonal numbers
+Then there is a metric given on <math>\mathbb{R}^3</math> as
-: <math>CD_n = CD_{n-1}+10(n-1) \, </math>
+:<math>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}
+</math>.
-where ''CD''<sub>1</sub> is 1.
+and we compute
-== Centered decagonal prime ==
+:<math>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^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2
+</math>
-A '''centered decagonal prime''' is a centered decagonal number that is [[prime number|prime]].  The first few centered decagonal primes are
+Therefore <math>g_\mathcal{C}=(m^2 a^2+2m^2ab\cos(n\cdot \tau)+m^2b^2\cos^2(n\cdot \tau)+b^2n^2)\mathrm{d}\tau\otimes \mathrm{d}\tau</math>
-: 11, 31, 61, 101, 151, 211, 281, 661, 911, 1051, 1201, 1361, 1531, 1901, 2311, 2531, 3001, 3251, 3511, 4651, 5281, ....
+==See also==
+*[[First fundamental form]]
-See also regular [[decagonal number]].
+[[Category:Differential geometry]]
-[[Category:Figurate numbers]]
+{{physics-stub}}
-[[ar:عدد ممركز عشاري]]
+[[it:Metrica indotta]]
-[[fr:Nombre décagonal centré]]
+[[ru:Индуцированная метрика]]
-[[ta:மையப்படுத்தப்பட்ட தசகோண எண்]]

Main Page: Difference between revisions

Revision as of 11:46, 13 August 2014

Example - Curve on a torus

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