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The Minkowski metric and Levi cita tensor is an invariant quantity in Euclidean flat space. But in curved space metric tensor varies. Analogous to it, is the Levi cita tensor varies in any Non-Euclidean space?

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On an arbitrary pseudo-Riemannian manidold, the Levi-Civita tensor is a pseudotensor while Levi-Civita symbol is a pseudotensor density. See e.g. this & this related Phys.SE posts.

For a generic coordinate system $x^{\mu}$, the corresponding Levi-Civita tensor components depend on $x^{\mu}$, if that's what you're asking.

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