Evaluating shear tensor by gravitational wave

The shear tensor is given by:

begin{equation}
sigma_{ij} = frac{1}{2}nabla_alpha U_beta + frac{1}{2}nabla_beta U_alpha + frac{1}{2} U_alpha U^mu nabla_mu U_beta + frac{1}{2} U_beta U^mu nabla_mu U_alpha + …
end{equation}

and the four velocity U is (1, 0, 0, 0). The covariant derivative is simply $Gamma^lambda_{alphabeta} U_lambda$. I tried the following.

begin{equation}
begin{split}
U_alpha U^mu nabla_mu U_beta &= g_{gammaalpha}g^{mugamma} U_mu U^mu nabla_mu U_beta
&= Gamma^lambda_{alphabeta} U_lambda
end{split}
end{equation}

where $g_{gammaalpha}g^{mugamma}$ becomes $delta^mu_alpha$ and applied to the covariant derivative, $U_mu U^mu$ becomes -1. If using this result, the first four terms cancel. However, the answer is simply summing up the free index $mu$ and the first four terms do not cancel.

The question is why the trick $U_alpha=g_{gammaalpha}g^{mugamma} U_mu$ is inconsistent with another approach of summing up.

Just to mention the Christoffel symbol not being zero here because of $h_{munu}$