Suppose we have a submanifold $M^n$ which is embedded in manifold $M^{n+2}$ and $g_{\mu \nu}$ denotes the metric of $M^{n+2}$. We know that the induced metric on the submanifold is defined by
\begin{equation} g_{ab}=\partial_a X^\mu \partial_b X^\nu g_{\mu\nu} \end{equation}
which in my opinion can be imagined as a projection of $g_{\mu\nu}$ onto the submanifold. Now by this metric and also using a normal vector on $M^n$ , we can calculate the intrinsic and extrinsic curvatures.
On the other hand, suppose we first find intrinsic curvature of $M^{n+2}$ and then by constructing
\begin{equation} h_{\mu\nu}=g_{\mu\nu}-n_\mu \cdot n_\nu \end{equation}
where $n_\mu$ is the normal vector, we project it onto the submanifold.
And the third case is we know the metric of $M^n$, which we call it $G_{ab}$, and by using it find the curvatures of $M^n$.
Now the question is whether the results of these three procedures are the same or not. By the way, am I right that $G_{a b}$ is different from $g_{ab}$?