Fubini–Study metric - Revision history

InternetArchiveBot: Rescuing 0 sources and tagging 1 as dead.) #IABot (v2.0.9.5

2024-05-13T23:28:21Z

Rescuing 0 sources and tagging 1 as dead.) #IABot (v2.0.9.5

← Previous revision		Revision as of 23:28, 13 May 2024
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	==Connection and curvature==		==Connection and curvature==
	The fact that the metric can be derived from the Kähler potential means that the [[Christoffel symbol]]s and the curvature tensors contain a lot of symmetries, and can be given a particularly simple form:<ref>Andrew J. Hanson, Ji-PingSha, "[ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf Visualizing the K3 Surface]" (2006)</ref> The Christoffel symbols, in the local affine coordinates, are given by		The fact that the metric can be derived from the Kähler potential means that the [[Christoffel symbol]]s and the curvature tensors contain a lot of symmetries, and can be given a particularly simple form:<ref>Andrew J. Hanson, Ji-PingSha, "[ftp://ftp.cs.indiana.edu/pub/hanson/forSha/AK3/old/K3-pix.pdf Visualizing the K3 Surface]{{Dead link\|date=May 2024 \|bot=InternetArchiveBot \|fix-attempted=yes }}" (2006)</ref> The Christoffel symbols, in the local affine coordinates, are given by
	:<math>		:<math>
	\Gamma^i_{\;jk}=g^{i\bar{m}}\frac{\partial g_{k\bar{m}}}{\partial z^j}		\Gamma^i_{\;jk}=g^{i\bar{m}}\frac{\partial g_{k\bar{m}}}{\partial z^j}

Wataxa: /* The n = 2 case */

2024-04-20T07:21:10Z

The n = 2 case

← Previous revision		Revision as of 07:21, 20 April 2024
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	z_1\bar{z}_1+z_2\bar{z}_2 &= r^2 = x^2+y^2+z^2+t^2 \\		z_1\bar{z}_1+z_2\bar{z}_2 &= r^2 = x^2+y^2+z^2+t^2 \\
	dz_1\,d\bar{z}_1 + dz_2\,d\bar{z}_2 &= dr^{\,2} + r^2(\sigma_1^{\,2}+\sigma_2^{\,2}+\sigma_3^{\,2}) \\		dz_1\,d\bar{z}_1 + dz_2\,d\bar{z}_2 &= dr^{\,2} + r^2(\sigma_1^{\,2}+\sigma_2^{\,2}+\sigma_3^{\,2}) \\
	~~\left(~~\bar{z}_1\,dz_1 + \bar{z}_2\,dz_2 ~~\right)^2~~ &= ~~r^2~~ ~~\left(dr^{~~\,~~2} +~~ r^2 \sigma_3~~^{\,2}~~ ~~\right)~~		\bar{z}_1\,dz_1 + \bar{z}_2\,dz_2 &= rdr + i\, r^2 \sigma_3
	\end{align}</math>		\end{align}</math>
	with the usual abbreviations that <math>dr^{\,2}=dr\otimes dr</math> and <math>\sigma_k^{\,2}=\sigma_k\otimes\sigma_k</math>.		with the usual abbreviations that <math>dr^{\,2}=dr\otimes dr</math> and <math>\sigma_k^{\,2}=\sigma_k\otimes\sigma_k</math>.
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	ds^2 &= \frac{dz_j\,d\bar{z}^j}{1+z_i\bar{z}^i}		ds^2 &= \frac{dz_j\,d\bar{z}^j}{1+z_i\bar{z}^i}
	- \frac{\bar{z}^j z_i\,dz_j\,d\bar{z}^i}{(1+z_i\bar{z}^i)^2} \\[5pt]		- \frac{\bar{z}^j z_i\,dz_j\,d\bar{z}^i}{(1+z_i\bar{z}^i)^2} \\[5pt]
	&= \frac{dr^2+r^2 (\sigma_1^2+\sigma_2^2+\sigma_3^2)}{1+r^2}		&= \frac{dr^{\,2}+r^2(\sigma_1^{\,2}+\sigma_2^{\,2}+\sigma_3^{\,2})}{1+r^2}
	- \frac{r^2 \~~left(dr^~~2 + r^2 \sigma_3^2 ~~\right)~~}{\left(1+r^2\right)^2} \\[4pt]		- \frac{r^2dr^{\,2} + r^4 \sigma_3^{\,2}}{\left(1+r^2\right)^2} \\[4pt]
	&= \frac{dr^2+r^2\sigma_3^2}{\left(1+r^2\right)^2} + \frac{r^2\left(\sigma_1^2+\sigma_2^2\right)}{1+r^2}		&= \frac{dr^{\,2}+r^2\sigma_3^{\,2}}{\left(1+r^2\right)^2} + \frac{r^2\left(\sigma_1^{\,2}+\sigma_2^{\,2}\right)}{1+r^2}
	\end{align}</math>		\end{align}</math>

Wataxa: /* In local affine coordinates */

2024-04-20T06:42:03Z

In local affine coordinates

← Previous revision		Revision as of 06:42, 20 April 2024
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	===In local affine coordinates===		===In local affine coordinates===
	Corresponding to a point in '''CP'''<sup>''n''</sup> with homogeneous coordinates ~~[''Z''~~<~~sub>0</sub~~>:~~...~~:~~''Z''<sub>''n''~~</~~sub~~>], there is a unique set of ''n'' coordinates ~~(''z''~~<~~sub>1</sub~~>,~~...~~,~~''z''<sub>''n''~~</~~sub~~>) such that		Corresponding to a point in '''CP'''<sup>''n''</sup> with homogeneous coordinates <math>[Z_0:\dots:Z_n] </math>, there is a unique set of ''n'' coordinates <math>(z_1,\dots,z_n)</math> such that
	:<math>[Z_0:\dots:Z_n] \sim [1,z_1,\dots,z_n],</math>		:<math>[Z_0:\dots:Z_n] \sim [1,z_1,\dots,z_n],</math>
	provided ~~''Z''~~<~~sub~~>0</~~sub~~>~~ ≠ 0~~; specifically, ~~''z''~~<~~sub~~>~~''j''</sub> ~~=~~ ''Z''<sub>''j''<~~/~~sub>/''Z''<sub>0~~</~~sub~~>. The ~~(''z''~~<~~sub>1</sub~~>,~~...~~,~~''z''<sub>''n''~~</~~sub~~>) form an [[affine coordinates\|affine coordinate system]] for '''CP'''<sup>''n''</sup> in the coordinate patch ~~''U''~~<~~sub~~>~~0</sub>~~ = ~~{''Z''<sub>~~0</~~sub~~>~~ ≠ 0}~~. One can develop an affine coordinate system in any of the coordinate patches ~~''U''~~<~~sub~~>~~''i''</sub> ~~=~~ ~~{~~''Z''<sub>''i''~~</~~sub~~>~~ ≠ 0}~~ by dividing instead by ~~''Z''~~<~~sub~~>~~''i''~~</~~sub~~> in the obvious manner. The ''n''+1 coordinate patches ~~''U''~~<~~sub~~>~~''i''~~</~~sub~~> cover '''CP'''<sup>''n''</sup>, and it is possible to give the metric explicitly in terms of the affine coordinates ~~(''z''~~<~~sub>1</sub~~>,~~...~~,~~''z''<sub>''n''~~</~~sub~~>) on ~~''U''~~<~~sub~~>~~''i''~~</~~sub~~>. The coordinate derivatives define a frame <math>\{\partial_1,\ldots,\partial_n\}</math> of the holomorphic tangent bundle of '''CP'''<sup>''n''</sup>, in terms of which the Fubini–Study metric has Hermitian components		provided <math>Z_0\neq 0</math>; specifically, <math>z_j=Z_j/Z_0</math>. The <math>(z_1,\dots,z_n)</math> form an [[affine coordinates\|affine coordinate system]] for '''CP'''<sup>''n''</sup> in the coordinate patch <math>U_0=\{Z_0\neq 0\}</math>. One can develop an affine coordinate system in any of the coordinate patches <math>U_i=\{Z_i\neq 0\}</math> by dividing instead by <math>Z_i</math> in the obvious manner. The ''n''+1 coordinate patches <math>U_i</math> cover '''CP'''<sup>''n''</sup>, and it is possible to give the metric explicitly in terms of the affine coordinates <math>(z_1,\dots,z_n)</math> on <math>U_i</math>. The coordinate derivatives define a frame <math>\{\partial_1,\ldots,\partial_n\}</math> of the holomorphic tangent bundle of '''CP'''<sup>''n''</sup>, in terms of which the Fubini–Study metric has Hermitian components

	:<math>g_{i\bar{j}} = h(\partial_i,\bar{\partial}_j) = \frac{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)\delta_{i\bar{j}} - \bar{z}_i z_j}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2}.</math>		:<math>g_{i\bar{j}} = h(\partial_i,\bar{\partial}_j) = \frac{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)\delta_{i\bar{j}} - \bar{z}_i z_j}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2}.</math>

Wataxa: /* The n = 2 case */

2024-04-20T03:59:29Z

The n = 2 case

← Previous revision		Revision as of 03:59, 20 April 2024
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	:<math>\begin{align}		:<math>\begin{align}
	z_1\bar{z}_1+z_2\bar{z}_2 &= r^2 = x^2+y^2+z^2+t^2 \\		z_1\bar{z}_1+z_2\bar{z}_2 &= r^2 = x^2+y^2+z^2+t^2 \\
	dz_1\,d\bar{z}_1 + dz_2\,d\bar{z}_2 &= dr^2 + r^2(\sigma_1^2+\sigma_2^2+\sigma_3^2) \\		dz_1\,d\bar{z}_1 + dz_2\,d\bar{z}_2 &= dr^{\,2} + r^2(\sigma_1^{\,2}+\sigma_2^{\,2}+\sigma_3^{\,2}) \\
	\left(\bar{z}_1\,dz_1 + \bar{z}_2\,dz_2 \right)^2 &= r^2 \left(dr^2 + r^2 \sigma_3^2 \right)		\left(\bar{z}_1\,dz_1 + \bar{z}_2\,dz_2 \right)^2 &= r^2 \left(dr^{\,2} + r^2 \sigma_3^{\,2} \right)
	\end{align}</math>		\end{align}</math>
	with the usual abbreviations that <math>dr^2=dr\otimes dr</math> and <math>\sigma_k^2=\sigma_k\otimes\sigma_k</math>.		with the usual abbreviations that <math>dr^{\,2}=dr\otimes dr</math> and <math>\sigma_k^{\,2}=\sigma_k\otimes\sigma_k</math>.

	The line element, starting with the previously given expression, is given by		The line element, starting with the previously given expression, is given by

Mgnbar: Undid revision 1216979422 by 137.189.49.242 (talk); explain why you think this is correct

2024-04-03T12:01:54Z

Undid revision 1216979422 by 137.189.49.242 (talk); explain why you think this is correct

← Previous revision		Revision as of 12:01, 3 April 2024
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	ds^2 &= g_{i\bar{j}} \, dz^i \, d\bar{z}^j \\[4pt]		ds^2 &= g_{i\bar{j}} \, dz^i \, d\bar{z}^j \\[4pt]
	&= \frac{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)\|d\mathbf{z}\|^2 - (\bar{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\bar{\mathbf{z}})}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2} \\[4pt]		&= \frac{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)\|d\mathbf{z}\|^2 - (\bar{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\bar{\mathbf{z}})}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2} \\[4pt]
	&= \frac{(1+z_i\bar{z}^i)\,dz_j\,d\bar{z}^j - \bar{z}^i ~~z_j~~\,~~dz_i~~\,d\bar{z}^j}{\left(1+z_i\bar{z}^i\right)^2}.		&= \frac{(1+z_i\bar{z}^i)\,dz_j\,d\bar{z}^j - \bar{z}^j z_i\,dz_j\,d\bar{z}^i}{\left(1+z_i\bar{z}^i\right)^2}.
	\end{align}		\end{align}
	</math>		</math>

137.189.49.242: /* In local affine coordinates */

2024-04-03T02:32:13Z

In local affine coordinates

← Previous revision		Revision as of 02:32, 3 April 2024
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	ds^2 &= g_{i\bar{j}} \, dz^i \, d\bar{z}^j \\[4pt]		ds^2 &= g_{i\bar{j}} \, dz^i \, d\bar{z}^j \\[4pt]
	&= \frac{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)\|d\mathbf{z}\|^2 - (\bar{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\bar{\mathbf{z}})}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2} \\[4pt]		&= \frac{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)\|d\mathbf{z}\|^2 - (\bar{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\bar{\mathbf{z}})}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2} \\[4pt]
	&= \frac{(1+z_i\bar{z}^i)\,dz_j\,d\bar{z}^j - \bar{z}^j ~~z_i~~\,~~dz_j~~\,d\bar{z}^i}{\left(1+z_i\bar{z}^i\right)^2}.		&= \frac{(1+z_i\bar{z}^i)\,dz_j\,d\bar{z}^j - \bar{z}^i z_j\,dz_i\,d\bar{z}^j}{\left(1+z_i\bar{z}^i\right)^2}.
	\end{align}		\end{align}
	</math>		</math>

BD2412: clean up spacing around commas and other punctuation fixes, replaced: ,W → , W, ,Y → , Y (3), ,Z → , Z (2), ,b → , b (6), ,d → , d (30), ,j → , j, ,k → , k, ,t → , t, ,y → , y, ,z → , z (3), , → , (2)

2024-01-22T15:23:30Z

clean up spacing around commas and other punctuation fixes, replaced: ,W → , W, ,Y → , Y (3), ,Z → , Z (2), ,b → , b (6), ,d → , d (30), ,j → , j, ,k → , k, ,t → , t, ,y → , y, ,z → , z (3), , → , (2)

← Previous revision		Revision as of 15:23, 22 January 2024
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	{{Short description\|Metric on a complex projective space endowed with Hermitian form}}		{{Short description\|Metric on a complex projective space endowed with Hermitian form}}
	In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'' , '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>		In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'', '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>

	A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a [[Unitary group\|unitary subgroup]] U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].		A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a [[Unitary group\|unitary subgroup]] U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].

Roffaduft at 06:52, 21 November 2023

2023-11-21T06:52:40Z

← Previous revision		Revision as of 06:52, 21 November 2023
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	In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'' , '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>		In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'' , '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>

	A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a unitary subgroup U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].		A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a [[Unitary group\|unitary subgroup]] U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].

	==Construction==		==Construction==

Roffaduft at 06:20, 21 November 2023

2023-11-21T06:20:47Z

← Previous revision		Revision as of 06:20, 21 November 2023
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	{{Short description\|Metric on a complex projective space endowed with Hermitian form}}		{{Short description\|Metric on a complex projective space endowed with Hermitian form}}
	In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] ~~on [[projective Hilbert space]], that is,~~ on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'' , '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>		In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'' , '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>

	A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a unitary subgroup U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].		A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a unitary subgroup U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].

Jacobolus: latex tweaks. in particular, slightly lower the superscript 2 in (|...|^2)^2, which results in smaller more legible parens

2023-11-15T07:46:00Z

latex tweaks. in particular, slightly lower the superscript 2 in (|...|^2)^2, which results in smaller more legible parens

← Previous revision		Revision as of 07:46, 15 November 2023
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	Specifically, one may define '''CP'''<sup>''n''</sup> to be the space consisting of all complex lines in '''C'''<sup>''n''+1</sup>, i.e., the quotient of '''C'''<sup>''n''+1</sup>\{0} by the [[equivalence relation]] relating all complex multiples of each point together. This agrees with the quotient by the diagonal [[Group action (mathematics)\|group action]] of the multiplicative group '''C'''<sup>*</sup> = '''C''' \ {0}:		Specifically, one may define '''CP'''<sup>''n''</sup> to be the space consisting of all complex lines in '''C'''<sup>''n''+1</sup>, i.e., the quotient of '''C'''<sup>''n''+1</sup>\{0} by the [[equivalence relation]] relating all complex multiples of each point together. This agrees with the quotient by the diagonal [[Group action (mathematics)\|group action]] of the multiplicative group '''C'''<sup>*</sup> = '''C''' \ {0}:

	:<math>\mathbf{CP}^n = \left\{ \mathbf{Z} = [Z_0,Z_1,\ldots,Z_n] \in {\mathbf C}^{n+1}\setminus\{0\}\, \right\} / \{ \mathbf{Z} \sim c\mathbf{Z}, c \in \mathbf{C}^* \}.</math>		:<math>\mathbf{CP}^n = \left\{ \mathbf{Z} = [Z_0,Z_1,\ldots,Z_n] \in {\mathbf C}^{n+1}\setminus\{0\} \right\} \big/ \{ \mathbf{Z} \sim c\mathbf{Z}, c \in \mathbf{C}^* \}.</math>

	This quotient realizes '''C'''<sup>''n''+1</sup>\{0} as a complex [[line bundle]] over the base space '''CP'''<sup>''n''</sup>. (In fact this is the so-called [[tautological bundle]] over '''CP'''<sup>''n''</sup>.) A point of '''CP'''<sup>''n''</sup> is thus identified with an equivalence class of (''n''+1)-tuples [''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>] modulo nonzero complex rescaling; the ''Z''<sub>''i''</sub> are called [[homogeneous coordinates]] of the point.		This quotient realizes '''C'''<sup>''n''+1</sup>\{0} as a complex [[line bundle]] over the base space '''CP'''<sup>''n''</sup>. (In fact this is the so-called [[tautological bundle]] over '''CP'''<sup>''n''</sup>.) A point of '''CP'''<sup>''n''</sup> is thus identified with an equivalence class of (''n''+1)-tuples [''Z''<sub>0</sub>,...,''Z''<sub>''n''</sub>] modulo nonzero complex rescaling; the ''Z''<sub>''i''</sub> are called [[homogeneous coordinates]] of the point.
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	Furthermore, one may realize this quotient mapping in two steps: since multiplication by a nonzero complex scalar ''z'' = ''R'' ''e''<sup>iθ</sup> can be uniquely thought of as the composition of a dilation by the modulus ''R'' followed by a counterclockwise rotation about the origin by an angle <math>\theta</math>, the quotient mapping '''C'''<sup>''n''+1</sup> → '''CP'''<sup>''n''</sup> splits into two pieces.		Furthermore, one may realize this quotient mapping in two steps: since multiplication by a nonzero complex scalar ''z'' = ''R'' ''e''<sup>iθ</sup> can be uniquely thought of as the composition of a dilation by the modulus ''R'' followed by a counterclockwise rotation about the origin by an angle <math>\theta</math>, the quotient mapping '''C'''<sup>''n''+1</sup> → '''CP'''<sup>''n''</sup> splits into two pieces.

	:<math>\mathbf{C}^{n+1}\setminus\{0\} \stackrel{(a)}\longrightarrow S^{2n+1} \stackrel{(b)}\longrightarrow \mathbf{CP}^n</math>		:<math>\mathbf{C}^{n+1}\setminus\{0\} \mathrel{\stackrel{(a)}\longrightarrow} S^{2n+1} \mathrel{\stackrel{(b)}\longrightarrow} \mathbf{CP}^n</math>

	where step (a) is a quotient by the dilation '''Z''' ~ ''R'''''Z''' for ''R'' ∈ '''R'''<sup>+</sup>, the multiplicative group of [[positive real numbers]], and step (b) is a quotient by the rotations '''Z''' ~ ''e''<sup>iθ</sup>'''Z'''.		where step (a) is a quotient by the dilation '''Z''' ~ ''R'''''Z''' for ''R'' ∈ '''R'''<sup>+</sup>, the multiplicative group of [[positive real numbers]], and step (b) is a quotient by the rotations '''Z''' ~ ''e''<sup>iθ</sup>'''Z'''.
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	provided ''Z''<sub>0</sub> ≠ 0; specifically, ''z''<sub>''j''</sub> = ''Z''<sub>''j''</sub>/''Z''<sub>0</sub>. The (''z''<sub>1</sub>,...,''z''<sub>''n''</sub>) form an [[affine coordinates\|affine coordinate system]] for '''CP'''<sup>''n''</sup> in the coordinate patch ''U''<sub>0</sub> = {''Z''<sub>0</sub> ≠ 0}. One can develop an affine coordinate system in any of the coordinate patches ''U''<sub>''i''</sub> = {''Z''<sub>''i''</sub> ≠ 0} by dividing instead by ''Z''<sub>''i''</sub> in the obvious manner. The ''n''+1 coordinate patches ''U''<sub>''i''</sub> cover '''CP'''<sup>''n''</sup>, and it is possible to give the metric explicitly in terms of the affine coordinates (''z''<sub>1</sub>,...,''z''<sub>''n''</sub>) on ''U''<sub>''i''</sub>. The coordinate derivatives define a frame <math>\{\partial_1,\ldots,\partial_n\}</math> of the holomorphic tangent bundle of '''CP'''<sup>''n''</sup>, in terms of which the Fubini–Study metric has Hermitian components		provided ''Z''<sub>0</sub> ≠ 0; specifically, ''z''<sub>''j''</sub> = ''Z''<sub>''j''</sub>/''Z''<sub>0</sub>. The (''z''<sub>1</sub>,...,''z''<sub>''n''</sub>) form an [[affine coordinates\|affine coordinate system]] for '''CP'''<sup>''n''</sup> in the coordinate patch ''U''<sub>0</sub> = {''Z''<sub>0</sub> ≠ 0}. One can develop an affine coordinate system in any of the coordinate patches ''U''<sub>''i''</sub> = {''Z''<sub>''i''</sub> ≠ 0} by dividing instead by ''Z''<sub>''i''</sub> in the obvious manner. The ''n''+1 coordinate patches ''U''<sub>''i''</sub> cover '''CP'''<sup>''n''</sup>, and it is possible to give the metric explicitly in terms of the affine coordinates (''z''<sub>1</sub>,...,''z''<sub>''n''</sub>) on ''U''<sub>''i''</sub>. The coordinate derivatives define a frame <math>\{\partial_1,\ldots,\partial_n\}</math> of the holomorphic tangent bundle of '''CP'''<sup>''n''</sup>, in terms of which the Fubini–Study metric has Hermitian components

	:<math>g_{i\bar{j}} = h(\partial_i,\bar{\partial}_j) = \frac{\left(1+\|\mathbf{z}\|^2\right)\delta_{i\bar{j}} - \bar{z}_i z_j}{\left(1+\|\mathbf{z}\|^2\right)^2}.</math>		:<math>g_{i\bar{j}} = h(\partial_i,\bar{\partial}_j) = \frac{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)\delta_{i\bar{j}} - \bar{z}_i z_j}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2}.</math>

	where \|'''z'''\|<sup>2</sup> = \|''z''<sub>1</sub>\|<sup>2</sup> + ... + \|''z''<sub>''n''</sub>\|<sup>2</sup>. That is, the [[Hermitian matrix]] of the Fubini–Study metric in this frame is		where \|'''z'''\|<sup>2</sup> = \|''z''<sub>1</sub>\|<sup>2</sup> + ... + \|''z''<sub>''n''</sub>\|<sup>2</sup>. That is, the [[Hermitian matrix]] of the Fubini–Study metric in this frame is

	:<math> \bigl[g_{i\bar{j}}\bigr] = \frac{1}{\left(1+\|\mathbf{z}\|^2\right)^2}		:<math> \bigl[g_{i\bar{j}}\bigr] = \frac{1}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2}
	\left[		\left[
	\begin{array}{cccc}		\begin{array}{cccc}
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	:<math>\begin{align}		:<math>\begin{align}
	ds^2 &= g_{i\bar{j}} \, dz^i \, d\bar{z}^j \\[4pt]		ds^2 &= g_{i\bar{j}} \, dz^i \, d\bar{z}^j \\[4pt]
	&= \frac{\left(1+\|\mathbf{z}\|^2\right)\|d\mathbf{z}\|^2 - (\bar{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\bar{\mathbf{z}})}{\left(1+\|\mathbf{z}\|^2\right)^2} \\[4pt]		&= \frac{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)\|d\mathbf{z}\|^2 - (\bar{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\bar{\mathbf{z}})}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2} \\[4pt]
	&= \frac{(1+z_i\bar{z}^i)\,dz_j\,d\bar{z}^j - \bar{z}^j z_i\,dz_j\,d\bar{z}^i}{\left(1+z_i\bar{z}^i\right)^2}.		&= \frac{(1+z_i\bar{z}^i)\,dz_j\,d\bar{z}^j - \bar{z}^j z_i\,dz_j\,d\bar{z}^i}{\left(1+z_i\bar{z}^i\right)^2}.
	\end{align}		\end{align}
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	Here the summation convention is used to sum over Greek indices α β ranging from 0 to ''n'', and in the last equality the standard notation for the skew part of a tensor is used:		Here the summation convention is used to sum over Greek indices α β ranging from 0 to ''n'', and in the last equality the standard notation for the skew part of a tensor is used:

	:<math>Z_{[\alpha}W_{\beta]} = \~~frac {1}{2}~~ \left(		:<math>Z_{[\alpha}W_{\beta]} = \tfrac12 \left(
	Z_{\alpha} W_{\beta} - Z_{\beta} W_{\alpha} \right).</math>		Z_{\alpha} W_{\beta} - Z_{\beta} W_{\alpha} \right).</math>

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	Namely, if ''z'' = ''x'' + i''y'' is the standard affine coordinate chart on the [[Riemann sphere]] '''CP'''<sup>1</sup> and ''x'' = ''r'' cos θ, ''y'' = ''r'' sin θ are polar coordinates on '''C''', then a routine computation shows		Namely, if ''z'' = ''x'' + i''y'' is the standard affine coordinate chart on the [[Riemann sphere]] '''CP'''<sup>1</sup> and ''x'' = ''r'' cos θ, ''y'' = ''r'' sin θ are polar coordinates on '''C''', then a routine computation shows

	:<math>ds^2= \frac{\operatorname{Re}(dz \otimes d\bar{z})}{\left(1+\|\mathbf{z}\|^2\right)^2}		:<math>ds^2= \frac{\operatorname{Re}(dz \otimes d\bar{z})}{\left(1+\|\mathbf{z}\|\vphantom{l}^2\right)^2}
	= \frac{dx^2+dy^2}{ \left(1+r^2\right)^2 }		= \frac{dx^2+dy^2}{ \left(1+r^2\right)^2 }
	= \~~frac{1}{4}~~(d\varphi^2 + \sin^2 \varphi\,d\theta^2)		= \tfrac14(d\varphi^2 + \sin^2 \varphi\,d\theta^2)
	= \~~frac{1}{4}~~ \, ds^2_{us}		= \tfrac14 \, ds^2_{us}
	</math>		</math>

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	where the curvature 2-form was expanded as a four-component tensor:		where the curvature 2-form was expanded as a four-component tensor:
	:<math>R^a_{\;\,b} = \~~frac{1}{2}~~R^a_{\;\,bcd}e^c\wedge e^d</math>		:<math>R^a_{\;\,b} = \tfrac12 R^a_{\;\,bcd}e^c\wedge e^d</math>

	The resulting [[Ricci tensor]] is constant		The resulting [[Ricci tensor]] is constant
	:<math>\operatorname{Ric}_{ab}=6\delta_{ab}</math>		:<math>\operatorname{Ric}_{ab}=6\delta_{ab}</math>
	so that the resulting [[Einstein equation]]		so that the resulting [[Einstein equation]]
	:<math>\operatorname{Ric}_{ab} - \~~frac{1}{2}~~\delta_{ab}R + \Lambda\delta_{ab} = 0</math>		:<math>\operatorname{Ric}_{ab} - \tfrac12 \delta_{ab}R + \Lambda\delta_{ab} = 0</math>
	can be solved with the [[cosmological constant]] <math>\Lambda=6</math>.		can be solved with the [[cosmological constant]] <math>\Lambda=6</math>.

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	:<math>W_{abcd}=R_{abcd} - 2\left(\delta_{ac}\delta_{bd} - \delta_{ad}\delta_{bc}\right)</math>		:<math>W_{abcd}=R_{abcd} - 2\left(\delta_{ac}\delta_{bd} - \delta_{ad}\delta_{bc}\right)</math>
	For the ''n'' = 2 case, the two-forms		For the ''n'' = 2 case, the two-forms
	:<math>W_{ab}=\~~frac{1}{2}~~W_{abcd} e^c \wedge e^d</math>		:<math>W_{ab}=\tfrac12 W_{abcd} e^c \wedge e^d</math>
	are self-dual:		are self-dual:

← Previous revision		Revision as of 15:23, 22 January 2024
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	{{Short description\|Metric on a complex projective space endowed with Hermitian form}}		{{Short description\|Metric on a complex projective space endowed with Hermitian form}}
	In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'' , '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>		In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'', '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>

	A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a [[Unitary group\|unitary subgroup]] U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].		A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a [[Unitary group\|unitary subgroup]] U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].

← Previous revision		Revision as of 06:20, 21 November 2023
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	{{Short description\|Metric on a complex projective space endowed with Hermitian form}}		{{Short description\|Metric on a complex projective space endowed with Hermitian form}}
	In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] ~~on [[projective Hilbert space]], that is,~~ on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'' , '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>		In [[mathematics]], the '''Fubini–Study metric''' (IPA: /fubini-ʃtuːdi/) is a [[Kähler metric]] on a [[complex projective space]] '''CP'''<sup>''n''</sup> endowed with a [[Hermitian form]]. This [[Metric (mathematics)\|metric]] was originally described in 1904 and 1905 by [[Guido Fubini]] and [[Eduard Study]].<ref>G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) ''Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti'' , '''63''' pp. 501–513</ref><ref>{{cite journal \| last=Study \| first=E. \| title=Kürzeste Wege im komplexen Gebiet \| journal=Mathematische Annalen \| publisher=Springer Science and Business Media LLC \| volume=60 \| issue=3 \| year=1905 \| issn=0025-5831 \| doi=10.1007/bf01457616 \| pages=321–378 \| s2cid=120961275 \| language=de}}</ref>

	A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a unitary subgroup U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].		A [[Hermitian form]] in (the vector space) '''C'''<sup>''n''+1</sup> defines a unitary subgroup U(''n''+1) in GL(''n''+1,'''C'''). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is [[homogeneous space\|homogeneous]]. Equipped with a Fubini–Study metric, '''CP'''<sup>''n''</sup> is a [[symmetric space]]. The particular normalization on the metric depends on the application. In [[Riemannian geometry]], one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the [[N-sphere\|(2''n''+1)-sphere]]. In [[algebraic geometry]], one uses a normalization making '''CP'''<sup>''n''</sup> a [[Hodge manifold]].