# compute the covariance derivative of the trace-free curvature

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 $$\Gamma_{ij}^k.$$