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

[center o bass]

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?

 

[lpetrich]

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