Conformal Killing vector field: Difference between revisions
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I put Riemannian metric instead of metric, with a link to metric tensors, and not to metric. Riemannian metric is not a metric, in the sense of metric spaces. |
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In [[conformal geometry]], the '''conformal Killing equation''' on a [[manifold]] of space-[[dimension]] ''n'' with [[ |
In [[conformal geometry]], the '''conformal Killing equation''' on a [[manifold]] of space-[[dimension]] ''n'' with [[Metric_tensor|Riemannian metric]] <math>g</math> describes those vector fields <math>X</math> which preserve <math>g</math> up to scale, i.e. |
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:<math>\mathcal{L}_{X}g = \lambda g</math> |
:<math>\mathcal{L}_{X}g = \lambda g</math> |
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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. |
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. |
Revision as of 09:52, 10 January 2019
In conformal geometry, the conformal Killing equation on a manifold of space-dimension n with Riemannian metric describes those vector fields which preserve up to scale, i.e.
for some function (where 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 . Therefore we can write the conformal Killing equation as
In abstract indices,
where the round brackets denote symmetrization.
For any n but 2, there is a finite number of solutions, specifying the conformal symmetry of that space, but in two dimensions, there is an infinity of solutions.
See also