I am trying to understand this question.
Essentially, given $n$D Riemannian space $(N,g)$ with an embedded codimension $1$ manifold $(M,\tilde{g})$ the goal is to characterize the mean curvature $H$ of $p\in M$. I'd like to do some parts a little more slowly. I'm a bit confused by some parts, so some help would be appreciated. :) I'll walk through and bold my questions.
First question: why can we say $\tilde{g}_{ij}=g_{ij}$? Is that just because we assume it (since $\tilde{g}$ is induced on $M$ by $g$) or is it a special property of $U$? For instance, if I perturb $M$, but not $N$, shouldn't its first fundamental form change (but not $N$'s)? From a local coordinates perspective, is $\tilde{g}$ just the upper $(n-1)\times(n-1)$ "submatrix" of $g$? (I feel this is likely a dumb question... sorry in advance.)
Second question: is my derivation for the mean curvature below correct?
By assumption, $\partial_n = \frac{\partial}{\partial x^n}\in (T_q M)^\perp$ such that $\vec{n}\cdot \partial_k=\delta_{kn}$, since we have codimension 1. Also, writing $\nabla_i \partial_j = \Gamma_{ij}^k\partial_k$.
Then, according to here, we can write the second fundamental form for a submanifold in a Riemannian space as \begin{align} \text{II}_{ij} &=\text{II}(\partial_i,\partial_j) = -\langle\nabla_i \vec{n},\partial_j\rangle = \langle \vec{n}, \nabla_i \partial_j\rangle = \langle \vec{n}, \Gamma_{ij}^k\partial_k\rangle = \langle \Gamma_{ij}^k\partial_k,\vec{n}\rangle \\ &= \Gamma_{ij}^k\langle \partial_k,\vec{n}\rangle = \Gamma_{ij}^k \delta_{nk}=\Gamma_{ij}^n \end{align} Then we get: $$ (n-1)H = \text{tr}(\tilde{g}^{-1} \text{II}) = \text{tr}(g^{ik} \Gamma^n_{kj}) = g^{ik} \Gamma^n_{ki} = g^{ik} \Gamma^n_{ik} $$ as the mean curvature by taking the trace of the shape operator. (Not sure if using Einstein summation was ok here)
The summations are a little weird: the indices of $\text{II}$ range from $1$ to $n-1$ (meaning $\text{II}$ is like the "upper submatrix" of $\Gamma_{ij}^n$), so the contraction $\tilde{g}^{ik}\text{II}_{kj} $ ranges up to $n-1$ as well. So:
Third Question: is $g^{ik}\Gamma^n_{ki}$ summed from $1$ to $n$? Everything else sums up to $n-1$ (as I suppose this does), but here these quantities naturally sum to $n$, so it is notationally confusing.