Why is there maximally 1/2 n(n+1) killing vectors?

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In summary, the maximum number of linearly independent killing vectors in an n-dimensional Riemannian manifold is n(n+1)/2. This fact is derived by first finding an equation for the second derivative of a Killing vector and then constraining the first derivative to be antisymmetric. This results in n(n-1)/2 possible values for the antisymmetrized first derivative, adding to a total of n(n+1)/2 possible values for ξ.
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center o bass
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It is often stated that there are maximum number of n/2 (n+1) linearly independent killing vectors in an n-dimensional Riemannian manifold.

How is this fact derived?
 
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I'd posted on that in the thread "Riemannian curvature of maximally symmetric spaces".

One first finds an equation for the second derivative of Killing vector ξ. This means that ξ anywhere is a function of ξ and its first derivative at some point. The defining equation for ξ constrains the first derivative of ξ to be antisymmetric or sort-of antisymmetric where ξ itself is 0. For n space dimensions:
  • Possible values of ξ: n
  • Possible values of the antisymmetrized first derivative of ξ: n(n-1)/2
Total: n(n+1)/2
 

1. What is the significance of having maximally 1/2 n(n+1) killing vectors?

The number of killing vectors in a given space determines the amount of symmetry present in that space. Having maximally 1/2 n(n+1) killing vectors is significant because it indicates that the space has the maximum amount of symmetry possible, making it highly symmetric and thus easier to study and understand.

2. How is the number of killing vectors related to the dimensionality of a space?

The number of killing vectors is directly related to the dimensionality of a space. For a space with n dimensions, there can exist at most 1/2 n(n+1) killing vectors. This means that the more dimensions a space has, the more killing vectors it can potentially have.

3. Why is the formula for the maximum number of killing vectors 1/2 n(n+1)?

The formula for the maximum number of killing vectors, 1/2 n(n+1), is derived from the Lie algebra of the rotation group in n dimensions. This group has n(n+1)/2 generators, and each generator corresponds to a killing vector. Therefore, the maximum number of killing vectors in a space is 1/2 n(n+1).

4. Are there cases where a space can have less than 1/2 n(n+1) killing vectors?

Yes, there are cases where a space can have less than 1/2 n(n+1) killing vectors. This occurs when the space has less symmetry, meaning it has fewer isometries or transformations that preserve distances and angles. For example, a sphere in 3 dimensions has only 3 killing vectors, even though the maximum number would be 1/2 (3)(3+1) = 6.

5. How do killing vectors relate to the concept of conserved quantities in physics?

Killing vectors play a crucial role in physics as they are related to the conserved quantities in a given space. These quantities include energy, momentum, and angular momentum, which are preserved in systems that possess certain symmetries. Killing vectors act as generators of these symmetries, allowing us to find and understand the conserved quantities in a system.

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