Let $F:M^n\to(N^{n+1},\bar{g})$ be a smooth immersion of a hypersurface in a Riemannian manifold. For simplicity, we assume both $M$ and $N$ are oriented closed manifolds. If $\nu$ denotes a unit normal vector field along the hypersurface, I would like to derive the Gauss-Weingarten equations in local coordinates $(\phi;x^i)$ around $p\in M$: $$\begin{align} \frac{\partial^2 F^\alpha}{\partial x^i\partial x^j}-\Gamma_{ij}^k\frac{\partial F^\alpha}{\partial x^k}+\overline{\Gamma}_{\beta\delta}^\alpha\frac{\partial F^\beta}{\partial x^i}\frac{\partial F^\delta}{\partial x^j}&=-h_{ij}\nu^\alpha,\tag{1}\\ \frac{\partial\nu^\alpha}{\partial x^i}+\overline{\Gamma}_{\beta\delta}^\alpha\frac{\partial F^\beta}{\partial x^i}\nu^\delta&=h_{ij}g^{j\ell}\frac{\partial F^\alpha}{\partial x^\ell}.\tag{2} \end{align}$$ $g$ is the induced metric on the hypersurface, and $h_{ij}$ gives the coordinate representation of the scalar second fundamental form $A$ in $(x^i)$. To derive the equations, we must have a solid knowledge of how the definition of $A$ works, but unfortunately I'm not quite aware of it. What the research paper told me is that $A(p)$ is a bilinear form $A(p):T_p M\times T_p M\to\Bbb{R}$ given by $$h_{ij}=\langle\overline{\nabla}_{e_i}\nu,e_j\rangle=-\langle\nu,\overline{\nabla}_{e_i}e_j\rangle\tag{3}$$ I was told $(e_i)$ is an orthonormal frame, but what do I do with it if I am to work with local coordinates $(x^i)$? Also, why isn't there any $F$-related terms in (3)? Can I possibly interpret $\overline{\nabla}_{e_i}e_j$ as $\frac{\partial^2 F}{\partial x^i\partial x^j}$, please?
My idea would be that because coordinate vector fields $\frac{\partial}{\partial x^i}$ form a local frame for $M$ and the differential $dF$ is injective, we can see that $\nu$ and $\frac{\partial F}{\partial x^i}\overset{\color{red}?}{:=}dF(\frac{\partial}{\partial x^i})$ form a basis pointwise on $N$. And that $F^\alpha$ should be the coordinate expression of $F$; to be precise, we should have $$\frac{\partial F}{\partial x^i}=\frac{\partial F^j}{\partial x^i}\frac{\partial}{\partial y^j},$$ where $y^j$'s are coordinates $(\psi;y^j)$ around $F(p)$ and $F^j$ is the $j$-th component of $\psi\circ F\circ\phi^{-1}$.
The meaning of the Gauss-Weingarten equations has a clear picture in differential geometry of curves and surfaces, but how do I generalize the derivation there to obtain the equations in question? Thank you.
Edit. The following are what equations (1) and (2) would be like in classical differential geometry (do Carmo, 2/e): these are nothing but linear combinations of the basis vectors with $N$ denoting a unit normal vector field along $S$. And I don't know why they become so ugly in an abstract setting.
$\newcommand\bp{\bar\partial}\newcommand\bn{\overline\nabla}\newcommand\bG{\overline\Gamma}$
ANOTHER WAY:
First, verify that each side of (1) and (2) transform under change of coordinates on both $M$ and $N$ like components of a tensor. It then suffices to verify the equations under a single choice of coordiinates. The trick now is to find a set of coordinates in a neighborhood of $p \in M$ and one in a neighborhood of $F(p) \in N$ that simplify the equations at $p$. I believe the right choice is to use geodesic normal coordinates centered at $p$ on $M$ and extend them to coordinates on $N$ using geodesics normal to $F(M)$. It's worth working this out carefully, too.
DIRECT APPROACH:
The second fundamental form is a symmetric $2$-tensor $h$. So if $$ h_{ij} = A(e_i,e_j) = \langle\overline\nabla_{e_i}\nu,e_j\rangle, $$ then given any two vectors $v = v^ie_i, w=w^je_j \in T_pM$, $$ A(v,w) = A(v^ie_i,w^je_j) = v^iw^jA(e_i,e_j) = v^iw^j\langle \overline\nabla_{e_i}\nu,e_j\rangle = \langle \overline\nabla_v\nu,w\rangle.$$
ADDED: But before we go any further, we should be more precise about this and also clarify what we mean by $\overline\nabla_v\nu$, since $v \in T_pM$ and $\nu(p) \in T_{F(p)}N$.
The connection acting on $\nu$ is the Levi-Civita connection on $N$ pulled back to the pullback bundle $F^*T_*N$, which is a vector bundle over $M$. In particular, given a section $\nu: M \rightarrow T_*N$ where $\nu(p) \in T_{F(p)}N$ and $v \in T_pM$, the covariant derivative of $\nu$ is defined to be $$\bn_v\nu(p) = \bn_{F_*v}\bar\nu(F(p)), $$ where $\bar\nu$ is the section of $T_*N$ restricted to $F(M)$ such that $$\nu = \bar\nu\circ F. $$ Also, the correct definition of $A$ is $$ A(v,w) = \langle \bn_v\nu,F_*w\rangle. $$
The calculations in coordinates start like this:
Let $\partial_1, \dots, \partial_n$ be the coordinate vector fields on $M$ and $\bp_1, \dots, \bp_{n+1}$ be the coordinate vector fields on $N$. Since $$ F_*\partial_i = \partial_iF^\beta\bp_\beta, $$ it follows that $$ \bn_{\partial_i}\bp_\beta = \bn_{\partial_iF^\alpha\bp_\alpha}\bp_\beta = \partial_iF^\alpha\bn_{\bp_\alpha}\bp_\beta = \partial_iF^\alpha\bG_{\alpha\beta}^\delta\bp_\delta. $$ Therefore, \begin{align*} h_{ij} &= A(\partial_i,\partial_j)\\ &= \langle\bn_{\partial_i}\nu,F_*\partial_j\rangle\\ &= \langle\bn_{\partial_i}(\nu^\beta\partial_\beta),\partial_jF^\delta\partial_\delta\rangle\\ &= \partial_jF^\delta\langle \partial_i\nu^\beta \partial_\beta+\nu^\gamma\bn_{\partial_i}\partial_\gamma,\partial_\delta\rangle\\ &= \partial_jF^\delta(\partial_i\nu^\beta + \nu^\gamma\partial_iF^\alpha\bG_{\alpha\gamma}^\beta)\langle\partial_\beta,\partial_\delta\rangle\\ &= \bar g_{\beta\delta}\partial_jF^\delta(\partial_i\nu^\beta + \nu^\gamma\partial_iF^\alpha\bG_{\alpha\gamma}^\beta). \end{align*} It follows that \begin{align*} g^{j\ell}\partial_\ell F^\alpha h_{ij} &=\partial_\ell F^\alpha g^{j\ell}\partial_jF^\delta \bar g_{\beta\delta}(\partial_i\nu^\beta + \nu^\gamma\partial_iF^\mu\bG_{\mu\gamma}^\beta). \end{align*} I beliieve the way to finish the calculation is to use the fact that $\bn_i\nu \in F_*T_pM$ and verify that the linear map \begin{align*} T_{F(p)}N &\rightarrow F_*T_pM\\ V^\beta\bp_\beta &\mapsto V^\beta\partial_\ell F^\alpha g^{j\ell}\partial_jF^\delta \bar g_{\beta\delta}\bp_\alpha \end{align*} is orthogonal projection.
The calculation for (1) is: \begin{align*} h_{ij} &= A(\partial_i,\partial_j)\\ &= \langle\nabla_{\partial_i}\nu,F_*\partial_j\rangle\\ &= \partial_i\langle\nu,F_*\partial_j\rangle - \langle \nu, \bn_i(F_*\partial_j)\rangle\\ &= -\langle\nu, \bn_{\partial_i}(\partial_jF^\alpha\bp_\alpha)\rangle\\ &= -\langle\nu, (\partial^2_{ij}F^\alpha +\partial_iF^\delta\partial_jF^\beta\bG_{\delta\beta}^\alpha)\bp_\alpha\rangle\\ &= -\langle\nu, (\partial^2_{ij}F^\alpha - \Gamma_{ij}^k\partial_kF^\alpha+\partial_iF^\delta\partial_jF^\beta\bG_{\delta\beta}^\alpha)\bp_\alpha\rangle\\ &= -\langle \nu, \nabla^2_{ij}F\rangle. \end{align*} Therefore, $$ \nabla^2F = A\nu. $$ In any codimension, the Hessian of $F$ is actually a bundle map $$\nabla^2F: S^2T_pM \rightarrow T^\perp_pM, $$ where $T^\perp_pM$ is the orthogonal complement of $F_*T_pM \subset T_{F(p)}N$. In codimension greater than $1$, there is no unique normal, so the second fundamental form is defined to be the bundle map $\nabla^2F$.