I saw in a talk that a surface gradient of $f:M \to \mathbb{R}$ where $M$ is a hypersurface in $\mathbb{R}^n$ defined as $\nabla_M f = \nabla f - (\nabla f \cdot N)N$ where $N$ is the unit normal vector on $M$ and $\nabla$ is the ordinary gradient.
I just started learning about the connection/covariant derivative on a manifold and am wondering about the link. Is the surface gradient as defined above just a choice of a particular connection? Does it have anything to do with the Levi-Civita connection?