Can we always choose an isometric slice chart for a submanifold of $\mathbb{R}^n$ Let $S$ be a submanifold of $\mathbb{R}^n$.
Let $p \in S$. Is there an isometric slice chart for $S$ in $\mathbb{R}^n$, around $p$?
i.e, I am asking whether there is a diffeomorphism $\phi$ from some open neighbourhood $U$, $p \in U\subseteq\mathbb{R}^n$ onto $V\subseteq\mathbb{R}^n$ such that:
$1) \, \phi(S\cap U) \, \text{ is a $k$-slice of  } \, \phi(U)=V$, that is
$\phi(S\cap U)=\{(x^1,...,x^k,x^{k+1},...,x^n) \in \phi(U)|x^{k+1}=c^{k+1}\dots x^n=c^n\}$ for some constants $c^1,...,c^n$.
2) $\phi:U \to \phi(U)$ is an isometry (when both $U,\phi(U)$ are endowed with the standard Euclidean metrics).
Update: As noted by Andrew D. Hwang, a necessary condition is that $S$ will be flat. However, I am not sure this is sufficient.
It is also immediate that $S$ must be totally geodesic. In fact $S$ must be even geodesically convex subset of $M$. (which implies totally geodesic, see here).
So, the question reduces to the following:
Can we choose a special chart as described above for every geodesically convex flat submanifold?
 A: $\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}$An isometric slice chart for $S$ exists near $p$ if and only if $S$ is affine in some neighborhood of $p$, in which case the slice chart is a rigid (Euclidean) motion: Let $U$ and $V$ be open balls in $\Reals^{n}$. If $\phi:U \to V$ is a diffeomorphism and an isometry of the Euclidean metric, then $\phi$ is the restriction of a rigid (Euclidean) motion.
Sketch: Without loss of generality, $\phi$ maps the origin to the origin. Because $\phi$ preserves norms (a.k.a. distance to the origin), $\phi$ preserves inner products by the polarization identity. Pick an orthogonal basis $(\Basis_{i})_{i=1}^{n}$ of equal-length vectors in $U$, use the fact $\phi$ preserves inner products to deduce
$$
\phi\left(\sum_{i=1}^{n} x^{i} \Basis_{i}\right)
  = \sum_{i=1}^{n} x^{i} \phi(\Basis_{i})
$$
for all $x = (x^{1}, \dots, x^{n})$ in some neighborhood of the origin, and conclude $\phi$ is the restriction of a linear mapping. (The case $n = 3$ is argued in more detail in O'Neill's Elementary Differential Geometry, Revised Second Edition, pp. 103-104.)
