Let $(M,g)$ be a $n$-dimensional Riemannian manifold and $\Sigma \subset B_r(x) \subset M$ be an embedded hypersurface of dimension $n-1$, with empty boundary and with a normalvector field $\nu$. Here $B_r(x) $ is a geodesic neighborhood of $x \in M$.
If now in addition $x \in \Sigma$ we can construct an orthonormal basis of $T_x M$ with $e_1 = \nu,e_2,...,e_n$. Let now be $\phi_x:B_r(x) \to \mathbb{R}$ be normal coordinate system in $x$ with respect to this basis. Then we get a hypersurface $\phi(\Sigma) \subset \mathbb{R}^{n}$.
Is then the vector $e_1=(1,0,...0)$ the normal vector of the surface $\phi(\Sigma) $ in the point $0=\phi(x) \in\phi(\Sigma) $ ?