I'm trying to derive the Gauss-Weingarten relations on a hypersurface immersed on an arbitrary Riemannian manifold (see page $468$ of this paper for the context):
$\begin{cases} \frac{\partial^2 F^{\alpha}}{\partial x_ix_j} - \Gamma^k_{ij} \frac{\partial F^{\alpha}}{\partial x_k} + \overline{\Gamma}^{\alpha}_{\rho\sigma} \frac{\partial F^{\rho}}{\partial x_i} \frac{\partial F^{\sigma}}{\partial x_j} = -h_{ij} \nu^{\alpha}\\ \frac{\partial \nu^{\alpha}}{\partial x_j} + \overline{\Gamma}^{\alpha}_{\rho\sigma} \frac{\partial F^{\rho}}{\partial x_j} \nu^{\sigma} = h_{jl} g^{lm} \frac{\partial F^{\alpha}}{\partial x_m} \end{cases}$
$\textbf{My attempt:}$
Firstly, I will prove the first equation. By one hand, I used the local formula for covariant derivative of the vector field $\overline{\nabla}_j F$ along $i$-parameter curves (see pages $122$ and $123$ of Barrett O'Neill - Semi-Riemannian Geometry with Applications to General Relativity for details):
$\begin{align*} \overline{\nabla}_i \overline{\nabla}_j F &= \sum_\limits{\alpha=1}^n \left\{ \frac{\partial^2 F^{\alpha}}{\partial x_i \partial x_j} + \sum_\limits{\rho,\sigma} \overline{\Gamma}^{\alpha}_{\rho\sigma} \frac{\partial F^{\rho}}{\partial x_i} \frac{\partial F^{\sigma}}{\partial x_j} \right\} \frac{\partial}{\partial x_{\alpha}} \end{align*}$
By the other hand,
$\begin{align*} \overline{\nabla}_i \overline{\nabla}_j F &= \sum_\limits{\alpha=0}^n \overline{\Gamma}^{\alpha}_{ij} \frac{\partial}{\partial x_{\alpha}}\\ &= \sum_\limits{k=1}^n \Gamma^k_{ij} \frac{\partial}{\partial x_k} - h_{ij} \nu \end{align*}$
Combining these two observations,
$$\frac{\partial^2 F^{\alpha}}{\partial x_ix_j} - \Gamma^k_{ij} \frac{\partial F^{\alpha}}{\partial x_k} + \overline{\Gamma}^{\alpha}_{\rho\sigma} \frac{\partial F^{\rho}}{\partial x_i} \frac{\partial F^{\sigma}}{\partial x_j} = -h_{ij} \nu,$$
which is different from the original equation by the term $-h_{ij} \nu$. I would like to know where I'm missing.
I will prove the second equation. By one hand, I used the local formula for covariant derivative of the vector field $\nu$ along $j$-parameter curves:
$\begin{align*} \overline{\nabla}_j \nu &= \sum_\limits{\alpha=0}^n \left\{ \frac{\nu^{\alpha}}{\partial x_j} + \sum_\limits{\rho,\sigma} \overline{\Gamma}^{\alpha}_{\rho\sigma} \frac{\partial F^{\rho}}{\partial x_j} \nu^{\sigma} \right\} \frac{\partial}{\partial x_{\alpha}} \end{align*}$
By the other hand,
$\begin{align*} \overline{\nabla}_j \nu &= h_{jl} g^{lm} \frac{\partial F^{\alpha}}{\partial x_m} \end{align*}$
I want to combine these two observations in order to obtain
$$\frac{\partial \nu^{\alpha}}{\partial x_j} + \overline{\Gamma}^{\alpha}_{\rho\sigma} \frac{\partial F^{\rho}}{\partial x_j} \nu^{\sigma} = h_{jl} g^{lm} \frac{\partial F^{\alpha}}{\partial x_m},$$
but I can't because I have a derivation $\frac{\partial}{\partial x_{\alpha}}$ by one hand and a number $\frac{\partial F^{\alpha}}{\partial x_m}$ by the other hand. I would like to know what I'm missing here.
Thanks in advance!
