Levi-Civita-connection of an embedded submanifold is induced by the orthogonal projection of the Levi-Civita-connection of the original manifold Let $(M, g)$ be a Riemannian manifold with Levi-Civita-connection $\nabla$, and let $N \subseteq M$ be an embedded submanifold with a $g$-induced Riemannian metric $h$. I now want to show that the Levi-Civita-connection $\tilde \nabla$ of $(N, h)$ is given by 
$$(\tilde \nabla_X Y)_p = \mathrm{pr}_{T_p M} (\nabla_{X_p} Z)_p\tag{$*$}$$
where 


*

*$\mathrm{pr}_{T_p M} : T_p M \to T_p N \subseteq T_p M$ is the $g$-orthogonal projection, and 

*$Z \in \mathfrak X(M)$ is a vector field on $M$, which is identical to $Y \in \mathfrak X(N)$ locally around $p \in N$.
Now I think this is mostly a matter of disentangling all the definitions and substituting them correctly wherever necessary, but I keep getting lost. Now my first problem is understanding why such a vector field $Z$ as desired even exists, and why the right-hand side of $(*)$ is independent of which $Z$ with this property I choose.
But even assuming that this is the case, where would I go from there? I thought about maybe using the Koszul formula or one of the basic properties of the Levi-Civita connection (because there's not much else that I know about it that might be helpful) but I'm not sure what exactly to do with them.

A possible way to  prove this is to remember that the LC connection is the unique torsion free connexion for which the metric tensor is parallel.
The fact that your formula gives a connexion is obvious. 
To check that it is torsion free, note that $g_N(\tilde \nabla _X Y, Z)= g(\nabla _X Y, Z)$ for every triple of tangent vector fields on $N$
To check that it preserves the induced tensor metric let $X,Y,Z$ three tangent vector fields on $N$. We can extend these fields on $M$ to compute :
$(\tilde \nabla _X g) (X,Y)= X. g(Y,Z)-g(\tilde \nabla _X Y, Z)- g(Y, \tilde \nabla _XZ)=X. g(Y,Z)-g( \nabla _X Y, Z)- g(Y,   \nabla _XZ) $
