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From the paper "Flow By Mean Curvature Of Convex Surfaces into Spheres," I did not understand the integration by parts on page $248,$ I am confused with the integration as follows:

$$\int \frac{2}{H^{\alpha}} \nabla_j h_{ij}^0 \nabla_i H d \mu.$$

Here, $\nabla_j$ represent the covariant derivative defined as $\frac{\partial}{\partial x_j}+\Gamma^{k}_{ij}$ and $h_{ij}^0$ is the traceless term defined as $h_{ij}-\frac{1}{n}g_{ij}.$

I am trying to compute $\nabla_j h^0_{ij}$ but do not have an idea because I do not know how to process the induced connection term $\Gamma_{ij}^k.$

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