[Differential Geometry] Simple relations between Killing vectors and curvature

[Kalma]

First of all, hello :)
I'd like to request some aid concerning a problem that is really getting to me. I know it should be simple but I'm not getting the right results.

Homework Statement

Given that $$V^{\mu}$$ is a Killing vector, prove that:

$$V^{\mu;\lambda}_{;\lambda} + R^{\mu}_{\lambda}V^{\lambda} = 0$$

And that:

$$V_{\lambda;\alpha\beta} = R_{\lambda\beta\alpha\mu}V^{\mu}$$

Where ";" represents the covariant derivative with respect to the indexes that follow it.

Homework Equations

The Killing Equation and the ones pertaining to the Riemann tensor is my best guess. But the problem is stated merely as I did in the previous section. There might be some "trick" I'm unaware of that does not use them.

The Attempt at a Solution

The first equation I hit a standstill almost instantly as I don't know how to handle the "contravariant derivative". Can I even define a "contravariant connection"? And how do I maintain the proper degress of freedom in the terms? The covariant derivative of something must still be a "something" of the same nature. I'm confused here, I admit.

On the second equation my work as proved more fruitful: I expand the two derivatives and obtain terms that resemble the Riemann tensor. My thought was to calculate the commutator to cut the extra terms and then use the fact that the cyclic sum of the last 3 indexes of the Riemann tensor vanishes together with the Killing equation to make the proof. But when I finish expanding the derivatives and subtract $$V_{\lambda;\beta\alpha}$$ from $$V_{\lambda;\alpha\beta}$$ I do not get a Riemann tensor multiplied by a Killing vector as I hoped I would, in fact, I get two terms on one component of the Killing vector, and two terms on another, different component of the Killing vector.

I hope I was clear. I do all my calculations by hand, and don't know how I can show them here, or even if you'd want to see them (I am very thorough). I think I didn't break any of the forum rules but if there are any I'm not aware of, please tell me so I can redo the post accordingly.

Thanks in advance for any help or insight. I know these should be easy, but I'm failing at a conceptual level, mainly I think I may be doing the second covariant derivative wrong, given the term from the first one :(

P.S.: Rewrote formulas using tex tags. It's much clearer now :)

Edit: removed the multitude of extra [1], [2] and [3] that appeared out of nowhere :S
Edit 2: Misspelled Riemann :) Double "n", one "m", must remember >_<

 

[NateD]

My answer:For the first equation, we will make use of the Killing equation, which states that V^{\mu}_{;\lambda} + V_{\lambda;\mu} = 0. Taking the covariant derivative of both sides of this equation with respect to the index \lambda gives us V^{\mu;\lambda}_{;\lambda} + V_{\lambda;\mu;\lambda} = 0. Then using the fact that the Riemann curvature tensor is antisymmetric in its last two indices, we can rewrite this as V^{\mu;\lambda}_{;\lambda} + R^{\mu}_{\lambda}V^{\lambda} = 0. For the second equation, we will use the fact that the Riemann curvature tensor is symmetric in its first two indices, so we have V_{\lambda;\alpha\beta} = V_{\lambda;\beta\alpha}. Taking the difference of both sides, we get V_{\lambda;\alpha\beta} - V_{\lambda;\beta\alpha} = 0. Using the Killing equation, this can be rewritten as V_{\lambda;\alpha\beta} - V_{\lambda;\beta\alpha} = R_{\lambda\beta\alpha\mu}V^{\mu}.