Killing Vector Fields: Generating Local Transformations?

In summary, the conversation discusses the possibility of a killing vector field generating a diffeomorphism that only shifts points inside a small part of the manifold and preserves points outside of it. It is suggested that this may not be possible due to the nature of isometries and geodesics.
  • #1
kakarukeys
190
0
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?
 
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  • #2
kakarukeys said:
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?

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)
 
  • #3
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?
 
  • #4
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.
 

1. What is a Killing vector field?

A Killing vector field is a type of vector field in differential geometry that preserves the metric structure of a manifold. In other words, it generates local transformations that preserve the distance and angles between points on the manifold.

2. How are Killing vector fields used in physics?

Killing vector fields are used in physics to study symmetries and conservation laws in physical systems. They are particularly useful in the study of general relativity, where they correspond to the conserved quantities of energy, momentum, and angular momentum.

3. Can Killing vector fields be used to solve differential equations?

Yes, Killing vector fields can be used to solve certain types of differential equations, specifically those related to symmetries and conservation laws. They can also be used to simplify the calculations involved in solving these equations.

4. How are Killing vector fields related to Lie groups?

Killing vector fields and Lie groups are closely related. In fact, the set of all Killing vector fields on a manifold forms a Lie algebra, which is a mathematical structure that is closely connected to Lie groups. This relationship is important in the study of symmetries in physics.

5. Are there any practical applications of Killing vector fields?

Yes, Killing vector fields have practical applications in various fields such as physics, engineering, and computer science. They are used in the development of mathematical models, simulations, and optimization algorithms. They also have applications in differential geometry, where they can be used to study the geometry of manifolds and map out their symmetries.

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