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space is flat it's invarainat under Poincare group - so in particular it's invariant under SO(3).

Spin operators are simply generators of SO(3). If the space isn't flat we cannot define

spin in this way. I know that in curved spacetime we should think of fileds rather than particles,

because notion of particle is not always well defined. So maybe better question is

how we define spin in globally hyberbolic and stationary spacetime?