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- About the properties of Killing Vector Fields for spherically symmetric manifold
A spherically symmetric manifold has, by definition, a set of 3 independent Killing Vector Fields (KVFs) satisfying: $$\begin{align}[R,S] &=T \nonumber \\ [S,T] &=R \nonumber \\ [T,R] &=S \nonumber \end{align}$$
These 3 KVFs define a linear subspace of the (infinite dimensional) vector space over ##\mathbb R## of the vector fields defined on the manifold ##M##. Indeed any linear combination with constant real coefficients of the 3 KVFs is a KVF and, from the properties of the commutator ##[~.,.]## it follows that the span of ##\{ T,R,S \}## closes w.r.t. it as well.
My question: taken 3 independent linear combinations of ##\{ T,R,S \}##, say ##\{X,Y,Z\}##, do they satisfy the three conditions above as well ?
These 3 KVFs define a linear subspace of the (infinite dimensional) vector space over ##\mathbb R## of the vector fields defined on the manifold ##M##. Indeed any linear combination with constant real coefficients of the 3 KVFs is a KVF and, from the properties of the commutator ##[~.,.]## it follows that the span of ##\{ T,R,S \}## closes w.r.t. it as well.
My question: taken 3 independent linear combinations of ##\{ T,R,S \}##, say ##\{X,Y,Z\}##, do they satisfy the three conditions above as well ?
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