Compute of exchange of covariant derivatives.

Question

Let $v=v_kdx^k$,$R_{ijkl}=g_{kp}R^p_{ijl}$,How to show that :

$$ \nabla_i\nabla_jv_k-\nabla_j\nabla_iv_k=R_{ijkl}g^{lm}v_m $$

Answer 1

Hint: Using the defintion of $R$, write $R_{ijk}^l$ in terms of $\Gamma_{ij}^k$ and its derivatives. Then your formula can be proved by direct computation, using

$$\nabla_i v_j = \partial_iv_j - \Gamma_{ij}^k v_k.$$