Let $(\tilde{M},\tilde{g})$ be an $(n-1)$-dimensional Riemannian manifold isometrically embedded in the $n$-dimensional $(M,g)$. The L-C connections are respectively $\tilde\nabla$ and $\nabla$. $N$ indicates the unit normal vector field to $\tilde{M}$. Then, for any pair of vector fields $X,Y$ tangent to $\tilde{M}$ one has the well-known: $$ \tilde\nabla_Y X = \nabla_Y X - g(\nabla_Y X,N)\,N \,, $$ stating that $\tilde\nabla$ is the projection of $\nabla$ into $T\tilde{M}$.
Does anyone know if this relation has a name?