Sorry if this is a too basic question, but I didn't find an answer anywhere:

The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\infty}(T^{\ast}M^{\otimes_{s}2})$ given by $$\Delta:=\mathrm{tr}(\nabla^{2}T)$$ Now, I am confused about the coordinate expression: Many references, for example this nice lecture notes by L. Nicolaescu, state that $$\Delta:=g^{ij}(\nabla_{i}\nabla_{j}+\Gamma_{ij}^{k}\nabla_{k})\tag{$\ast$}$$ where $\nabla$ denotes the Levi-Civita connection. However, I am a bit puzzled by the second term. For example, lets say I'll take the specific case $k=0$. Then, using formula ($\ast$) I get $$\Delta=g^{ij}(\underbrace{\nabla_{i}\nabla_{j}f}_{\partial_{i}\partial_{j}f-\Gamma_{ij}^{k}\nabla_{k}f}+\Gamma_{ij}^{k}\nabla_{k}f)=g^{ij}\partial_{i}\partial_{j}f$$ which is wrong, since we should obtain the Laplace-Beltrami operator $$\Delta_{\mathrm{LB}}f=g^{ij}(\partial_{i}\partial_{j}f-\Gamma_{ij}^{k}\partial_{k}f)$$

I assume it is just a missunderstanding of notation or terminology. Any help appreciated.