Answer to Question #283318 in Differential Geometry | Topology for Sabiha

Question #283318

Find killing equations for spherical polar coordinates


1
Expert's answer
2021-12-29T16:30:21-0500

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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