Namespaces
Variants
Actions

Difference between revisions of "Weyl tensor"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (AUTOMATIC EDIT (latexlist): Replaced 70 formulas out of 81 by TEX code with an average confidence of 2.0 and a minimal confidence of 2.0.)
Line 1: Line 1:
In [[Riemannian geometry|Riemannian geometry]] one has a [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201201.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201202.png" /> which admits a [[Metric tensor|metric tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201203.png" /> whose signature is arbitrary. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201204.png" /> be the unique [[Levi-Civita connection|Levi-Civita connection]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201205.png" /> arising from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201206.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201207.png" /> be the associated [[Curvature tensor|curvature tensor]] with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201208.png" />. Of importance in Riemannian geometry is the idea of a conformal change of metric, that is, the replacement of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201209.png" /> by the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012010.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012011.png" /> is a nowhere-zero real-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012012.png" />. The metrics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012014.png" /> are then said to be conformally related (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012015.png" /> is said to be  "conformal"  to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012016.png" />). One now asks for the existence of a tensor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012017.png" /> which is constructed from the original metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012018.png" /> and which would be unchanged if it were to be replaced with another metric conformally related to it. (It is noted here that the curvature tensor would only be unaffected by such a change, in general, if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012019.png" /> were constant.) The answer was provided mainly by H. Weyl [[#References|[a1]]], but with important contributions from J.A. Schouten [[#References|[a2]]] (see also [[#References|[a3]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012020.png" />, Weyl constructed the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012021.png" /> (now called the Weyl tensor) with components given by
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,  
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
Out of 81 formulas, 70 were replaced by TEX code.-->
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012023.png" /></td> </tr></table>
+
{{TEX|semi-auto}}{{TEX|partial}}
 +
In [[Riemannian geometry|Riemannian geometry]] one has a [[Manifold|manifold]] $M$ of dimension $n$ which admits a [[Metric tensor|metric tensor]] $g$ whose signature is arbitrary. Let $\Gamma$ be the unique [[Levi-Civita connection|Levi-Civita connection]] on $M$ arising from $g$ and let $\mathcal{R}$ be the associated [[Curvature tensor|curvature tensor]] with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w1201208.png"/>. Of importance in Riemannian geometry is the idea of a conformal change of metric, that is, the replacement of the metric $g$ by the metric $\phi g$ where $\phi$ is a nowhere-zero real-valued function on $M$. The metrics $g$ and $g ^ { \prime }$ are then said to be conformally related (or $g ^ { \prime }$ is said to be  "conformal"  to $g$). One now asks for the existence of a tensor on $M$ which is constructed from the original metric on $M$ and which would be unchanged if it were to be replaced with another metric conformally related to it. (It is noted here that the curvature tensor would only be unaffected by such a change, in general, if the function $\phi$ were constant.) The answer was provided mainly by H. Weyl [[#References|[a1]]], but with important contributions from J.A. Schouten [[#References|[a2]]] (see also [[#References|[a3]]]). For $n &gt; 3$, Weyl constructed the tensor $C$ (now called the Weyl tensor) with components given by
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012024.png" /> are the [[Ricci tensor|Ricci tensor]] components, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012025.png" /> is the Ricci scalar and square brackets denote the usual skew-symmetrization of indices. If this tensor is written out in terms of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012026.png" /> and its first- and second-order derivatives, it can then be shown to be unchanged if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012027.png" /> is replaced by the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012028.png" />. (It should be noted that this is not true of the tensor with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012029.png" />, which would be scaled by a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012030.png" /> on exchanging <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012032.png" />.) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012033.png" /> is a flat metric (so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012034.png" />), then the Weyl tensor constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012035.png" /> (and from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012036.png" />) is zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012037.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012038.png" /> gives rise, from (a1), to a zero Weyl tensor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012039.png" />, then for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012041.png" /> there are a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012042.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012044.png" />, a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012045.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012046.png" /> and a flat metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012048.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012050.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012051.png" /> is locally conformal to a flat metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012052.png" />). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012053.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012054.png" />, the latter is called conformally flat.
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012022.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a1)</td></tr></table>
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012055.png" />, it can be shown from (a1) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012056.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012057.png" />. Since not every metric on such a manifold is locally conformally related to a flat metric, the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012058.png" /> is no longer appropriate. The situation was resolved by Schouten [[#References|[a2]]] when he found that the tensor given in components by
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012023.png"/></td> </tr></table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012059.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
where $R _ { ab } \equiv R ^ { c } \square _ { a c b }$ are the [[Ricci tensor|Ricci tensor]] components, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012025.png"/> is the Ricci scalar and square brackets denote the usual skew-symmetrization of indices. If this tensor is written out in terms of the metric $g$ and its first- and second-order derivatives, it can then be shown to be unchanged if $g$ is replaced by the metric $g ^ { \prime }$. (It should be noted that this is not true of the tensor with components $C_{abcd}$, which would be scaled by a factor $\phi$ on exchanging $g$ for $g ^ { \prime }$.) If $g$ is a flat metric (so that $\mathcal{R} = 0$), then the Weyl tensor constructed from $g$ (and from $g ^ { \prime }$) is zero on $M$. Conversely, if $g$ gives rise, from (a1), to a zero Weyl tensor on $M$, then for each $p$ in $M$ there are a neighbourhood $U$ of $p$ in $M$, a real-valued function $\psi$ on $U$ and a flat metric $h$ on $U$ such that $g = \psi h$ on $U$ (i.e. $g$ is locally conformal to a flat metric on $M$). When $C = 0$ on $M$, the latter is called conformally flat.
  
(using a semi-colon to denote a [[Covariant derivative|covariant derivative]] with respect to the [[Levi-Civita connection|Levi-Civita connection]] arising from the metric) played exactly the same role in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012060.png" /> as did <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012061.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012062.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012063.png" />, every metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012064.png" /> is locally conformally related to a flat metric [[#References|[a3]]].
+
If $n = 3$, it can be shown from (a1) that $C \equiv 0$ on $M$. Since not every metric on such a manifold is locally conformally related to a flat metric, the tensor $C$ is no longer appropriate. The situation was resolved by Schouten [[#References|[a2]]] when he found that the tensor given in components by
  
The tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012065.png" /> has all the usual algebraic symmetries of the [[Curvature tensor|curvature tensor]], together with the extra relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012066.png" />. If the Ricci tensor is zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012067.png" />, the Weyl tensor and the curvature tensor are equal on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012068.png" />. The tensor introduced in (a2) by Schouten possesses the algebraic identities
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012059.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a2)</td></tr></table>
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
(using a semi-colon to denote a [[Covariant derivative|covariant derivative]] with respect to the [[Levi-Civita connection|Levi-Civita connection]] arising from the metric) played exactly the same role in dimension $3$ as did $C$ for $n &gt; 3$. If $n = 2$, every metric on $M$ is locally conformally related to a flat metric [[#References|[a3]]].
  
It is interesting to ask if two metrics on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012070.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012071.png" />) having the same Weyl tensor as in (a1) are necessarily (locally) conformally related. The answer is clearly no if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012072.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012073.png" /> is not zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012074.png" />, the answer is still no, a counter-example (at least) being available for a space-time manifold (i.e. a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012075.png" />-dimensional manifold admitting a metric with Lorentz signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012076.png" />).
+
The tensor $C$ has all the usual algebraic symmetries of the [[Curvature tensor|curvature tensor]], together with the extra relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012066.png"/>. If the Ricci tensor is zero on $M$, the Weyl tensor and the curvature tensor are equal on $M$. The tensor introduced in (a2) by Schouten possesses the algebraic identities
 +
 
 +
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012069.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a3)</td></tr></table>
 +
 
 +
It is interesting to ask if two metrics on $M$ ($n &gt; 3$) having the same Weyl tensor as in (a1) are necessarily (locally) conformally related. The answer is clearly no if $C = 0$. If $C$ is not zero on $M$, the answer is still no, a counter-example (at least) being available for a space-time manifold (i.e. a $4$-dimensional manifold admitting a metric with Lorentz signature $( + + + - )$).
  
 
The Weyl tensor finds many uses in [[Differential geometry|differential geometry]] and also in Einstein's general [[Relativity theory|relativity theory]]. In the latter it has important physical interpretations and its algebraic classification is the famous [[Petrov classification|Petrov classification]] of gravitational fields [[#References|[a4]]]
 
The Weyl tensor finds many uses in [[Differential geometry|differential geometry]] and also in Einstein's general [[Relativity theory|relativity theory]]. In the latter it has important physical interpretations and its algebraic classification is the famous [[Petrov classification|Petrov classification]] of gravitational fields [[#References|[a4]]]
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Weyl,  "Reine Infinitesimalgeometrie"  ''Math. Z.'' , '''2'''  (1918)  pp. 384–411</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.A. Schouten,  "Ueber die konforme Abbildung <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w120/w120120/w12012077.png" />-dimensionaler Mannigfaltigkeiten mit quadratischer Massbestimmung auf eine Mannigfaltigkeit mit Euklidischer Massbestimmung"  ''Math. Z.'' , '''11'''  (1921)  pp. 58–88</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1966)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.Z. Petrov.,  "Einstein spaces" , Pergamon  (1969)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Weyl,  "Reine Infinitesimalgeometrie"  ''Math. Z.'' , '''2'''  (1918)  pp. 384–411</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J.A. Schouten,  "Ueber die konforme Abbildung $n$-dimensionaler Mannigfaltigkeiten mit quadratischer Massbestimmung auf eine Mannigfaltigkeit mit Euklidischer Massbestimmung"  ''Math. Z.'' , '''11'''  (1921)  pp. 58–88</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1966)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A.Z. Petrov.,  "Einstein spaces" , Pergamon  (1969)</td></tr></table>

Revision as of 16:56, 1 July 2020

In Riemannian geometry one has a manifold $M$ of dimension $n$ which admits a metric tensor $g$ whose signature is arbitrary. Let $\Gamma$ be the unique Levi-Civita connection on $M$ arising from $g$ and let $\mathcal{R}$ be the associated curvature tensor with components . Of importance in Riemannian geometry is the idea of a conformal change of metric, that is, the replacement of the metric $g$ by the metric $\phi g$ where $\phi$ is a nowhere-zero real-valued function on $M$. The metrics $g$ and $g ^ { \prime }$ are then said to be conformally related (or $g ^ { \prime }$ is said to be "conformal" to $g$). One now asks for the existence of a tensor on $M$ which is constructed from the original metric on $M$ and which would be unchanged if it were to be replaced with another metric conformally related to it. (It is noted here that the curvature tensor would only be unaffected by such a change, in general, if the function $\phi$ were constant.) The answer was provided mainly by H. Weyl [a1], but with important contributions from J.A. Schouten [a2] (see also [a3]). For $n > 3$, Weyl constructed the tensor $C$ (now called the Weyl tensor) with components given by

(a1)

where $R _ { ab } \equiv R ^ { c } \square _ { a c b }$ are the Ricci tensor components, is the Ricci scalar and square brackets denote the usual skew-symmetrization of indices. If this tensor is written out in terms of the metric $g$ and its first- and second-order derivatives, it can then be shown to be unchanged if $g$ is replaced by the metric $g ^ { \prime }$. (It should be noted that this is not true of the tensor with components $C_{abcd}$, which would be scaled by a factor $\phi$ on exchanging $g$ for $g ^ { \prime }$.) If $g$ is a flat metric (so that $\mathcal{R} = 0$), then the Weyl tensor constructed from $g$ (and from $g ^ { \prime }$) is zero on $M$. Conversely, if $g$ gives rise, from (a1), to a zero Weyl tensor on $M$, then for each $p$ in $M$ there are a neighbourhood $U$ of $p$ in $M$, a real-valued function $\psi$ on $U$ and a flat metric $h$ on $U$ such that $g = \psi h$ on $U$ (i.e. $g$ is locally conformal to a flat metric on $M$). When $C = 0$ on $M$, the latter is called conformally flat.

If $n = 3$, it can be shown from (a1) that $C \equiv 0$ on $M$. Since not every metric on such a manifold is locally conformally related to a flat metric, the tensor $C$ is no longer appropriate. The situation was resolved by Schouten [a2] when he found that the tensor given in components by

(a2)

(using a semi-colon to denote a covariant derivative with respect to the Levi-Civita connection arising from the metric) played exactly the same role in dimension $3$ as did $C$ for $n > 3$. If $n = 2$, every metric on $M$ is locally conformally related to a flat metric [a3].

The tensor $C$ has all the usual algebraic symmetries of the curvature tensor, together with the extra relation . If the Ricci tensor is zero on $M$, the Weyl tensor and the curvature tensor are equal on $M$. The tensor introduced in (a2) by Schouten possesses the algebraic identities

(a3)

It is interesting to ask if two metrics on $M$ ($n > 3$) having the same Weyl tensor as in (a1) are necessarily (locally) conformally related. The answer is clearly no if $C = 0$. If $C$ is not zero on $M$, the answer is still no, a counter-example (at least) being available for a space-time manifold (i.e. a $4$-dimensional manifold admitting a metric with Lorentz signature $( + + + - )$).

The Weyl tensor finds many uses in differential geometry and also in Einstein's general relativity theory. In the latter it has important physical interpretations and its algebraic classification is the famous Petrov classification of gravitational fields [a4]

References

[a1] H. Weyl, "Reine Infinitesimalgeometrie" Math. Z. , 2 (1918) pp. 384–411
[a2] J.A. Schouten, "Ueber die konforme Abbildung $n$-dimensionaler Mannigfaltigkeiten mit quadratischer Massbestimmung auf eine Mannigfaltigkeit mit Euklidischer Massbestimmung" Math. Z. , 11 (1921) pp. 58–88
[a3] L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1966)
[a4] A.Z. Petrov., "Einstein spaces" , Pergamon (1969)
How to Cite This Entry:
Weyl tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_tensor&oldid=50152
This article was adapted from an original article by G.S. Hall (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article