# 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.

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 :
$(\tilde \nabla _X g) (X,Y)= X. g(Y,Z)-g(\tilde \nabla _X Y, Z)- g(Y, \tilde \nabla _XZ)=X. g(Y,Z)-g( \nabla _X Y, Z)- g(Y, \nabla _XZ)$
• Why $g_N(\tilde{\nabla}_XY,Z)=g(\nabla_X Y,Z)$ and how does that implie that it is torsion free? – Andre Gomes May 14 at 4:45
• The identity is just linear algebra, as ${\tilde \nabla} _X Y-\nabla _X Y$ is orthogonal to $Z$ by definition. For torsion free, consider two vector fields tangent to $N$ and compute $T(X,Y)=\ \nabla _X Y- \nabla _Y X-[X,Y]$. Then project orthogonaly on $N$ and you get the result. – Thomas May 15 at 5:30