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Killing vector fields

  1. 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?
     
  2. jcsd
  3. by definition, killing vector fields generate isometries. so, for a global killing v.f. to exist, just find an isometry that fixes the entire manifold except for a "small part" of it. (I'm a bit skeptical about being able to do that, myself)
     
  4. Tricky question... I think the answer is "no", at least for a connected manifold.

    On a "small" part of the manifold, we may as well take the metric to be flat -- so, Minkowski or Euclidean. In n dimensions, flat space has n(n+1)/2 Killing vector fields. Any Killing vector field is a linear superposition of these. And I don't think you can get any linear superposition of these to vanish outside an enclosed region -- which is what you're asking for.

    I'm sure there's an elegant way to prove this -- I'd have to think about it more. Maybe someone else will chime in?
     
  5. I'm thinking if this is a proof:

    An isometry preserves the metric, it therefore preserves geodesics. It maps geodesics to geodesics. Assuming the isometry is identity outside a small region, choose three points that are on a geodesics, two outside and one inside the region. The isometry keeps the two points and any point between them outside the region fixed but shifts the 3rd points. So we have two distinct geodesics connecting two points, which is not allowed by ODE's theory, the geodesics equation is a simple ODE that has a unique solution given an initial point and initial velocity.
     
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