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## Main Question or Discussion Point

Can a killing vector field generate a diffeomorphism that only shifts points inside a small part of the manifold and preserves points outside of it?

Rotation preserves the metric of a sphere but shifts every points on the sphere, I'd to find out if there is a killing vector field that generates local transformation?

Rotation preserves the metric of a sphere but shifts every points on the sphere, I'd to find out if there is a killing vector field that generates local transformation?