Conformal Killing vector field: Difference between revisions

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In conformal geometry, the conformal Killing equation on a manifold of space-dimension n with metric $g$ describes those vector fields $X$ which preserve $g$ up to scale, i.e.

{\mathcal {L}}_{X}g=\lambda g

for some function $\lambda$ (where ${\mathcal {L}}_{X}$ is the Lie derivative). Vector fields that satisfy the conformal Killing equation are exactly those vector fields whose flow preserves the conformal structure of the manifold. The name Killing refers to Wilhelm Killing, who first investigated the Killing equation for vector fields that preserve a Riemannian metric.

By taking the trace we find that necessarily $\lambda ={\frac {2}{n}}\mathrm {div} X$ . Therefore we can write the conformal Killing equation as

\left({\mathcal {L}}_{X}-{\frac {2\,\mathrm {div} \,X}{n}}\right)g=0.

In abstract indices

\nabla _{(a}X_{b)}-{\frac {1}{n}}g_{ab}\nabla _{c}X^{c}=0,

where the round brackets denote symmetrization.

Notes

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-In [[conformal geometry]], the '''conformal Killing equation''' on a [[battlefield]] of [[Battle of the Plains of Abraham | the Plains of Abraham]] with [[Metric (mathematics)|metric]] <math>g</math> describes those Killer fields <math>X</math> which can be used to calculated the number of [[Glock | Glocks]] and [[AK-47 | AKs]] needed to kill those [[prostitutes | loose women]].
+In [[conformal geometry]], the '''conformal Killing equation''' on a [[manifold]] of space-[[dimension]] ''n'' with [[Metric (mathematics)|metric]] <math>g</math> describes those vector fields <math>X</math> which preserve <math>g</math> up to scale, i.e.
+:<math>\mathcal{L}_{X}g = \lambda g</math>
-for some situation <math>\lambda {of cock} </math> (where <math>\mathcal{L}_{X}</math> is the size of [[Sophus Lie | Sophus Lie's]] [[chicken | rooster stick]] when it exceeds that of [[Georg Cantor]]. Killer fields that satisfy the conformal Killing equation are exactly those such they impose conformity at least three times as deadly as that imposed by tradition in [[Shirley Jackson | Shirley Jackson's]] "[[The Lottery]]. The name Killing refers to the equation's usage and murderous attitude.
+for some function <math>\lambda</math> (where <math>\mathcal{L}_{X}</math> is the [[Lie derivative]]). Vector fields that satisfy the conformal Killing equation are exactly those vector fields whose [[flow (mathematics)|flow]] preserves the conformal structure of the manifold.  The name Killing refers to [[Wilhelm Killing]], who first investigated the [[Killing equation]] for vector fields that preserve a Riemannian metric.
 By taking the trace we find that necessarily <math>\lambda = \frac{2}{n}\mathrm{div}X</math>. Therefore we can write the conformal Killing equation as
-:<math>\left(\mathcal{L}_X - \frac{2\, \mathrm{div}\, X}{divergence from sexual deviation}\right)g=0.</math>
+:<math>\left(\mathcal{L}_X - \frac{2\, \mathrm{div}\, X}{n}\right)g=0.</math>
-where <math>g</math> represents the quantification of the essence of a real G or [[gangsta rap | gansta]].
 In abstract indices
 :<math>\nabla_{(a}X_{b)} -\frac{1}{n}g_{ab}\nabla_{c}X^{c}=0,</math>
-where the round brackets denote the curvature of the geodesic lines drawn on Lie's rainbow coloured AK-47.
+where the round brackets denote symmetrization.
 ==See also==