Find killing equations for spherical polar coordinates
Killing vector field is a vector field on a Riemannian manifold that preserves the metric.
Killing equation:
"\\nabla_{\\mu}X_{v}+\\nabla_{v}X_{\\mu}=0"
Killing fields on a 2-sphere:
The conventional metric on the sphere is
"g(\\Omega)=d\\theta^2+sin^2\\theta d\\phi^2"
Rotation about the pole should be an isometry. The infinitesimal generator of a rotation can then be identified as a generator of the Killing field. This can be
"U=\\partial_ \\phi"
The surface of the sphere is two-dimensional, and there is another generator of isometries; it can be taken as
"V=\\partial_ \\theta"
conventional 3-space coordinate system is given by
"x=sin\\theta,y=sin\\theta sin \\phi,z=cos\\theta"
first generator, rotation about the z-axis:
"R=x\\partial _y-y\\partial_x=sin^2\\theta \\partial \\phi"
second generator, rotation about the x-axis:
"S=z\\partial _y-y\\partial_z"
third generator, rotation about the x-axis:
"T=z\\partial _x-x\\partial_z"
Normalizing this, and expressing these in spherical coordinates:
"R'=\\frac{R}{sin^2\\theta}=\\partial_ \\phi"
"S'=\\frac{S}{sin^2\\theta}=sin\\phi \\partial_ \\theta +cot\\theta cos\\phi \\partial _\\phi"
"T'=\\frac{T}{sin^2\\theta}=cos\\phi \\partial_ \\theta +cot\\theta sin\\phi \\partial _\\phi"
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