Riemannian gradient in the case of an embedded manifold

Question

I am reading a paper regarding optimization on matrix manifolds. In this paper, they state that "The Riemannian gradient in the case of an embedded manifold" is simply the projection on the tangent space of the Euclidean gradient $grad \ f$". I would like to know if this is stated as a proposition/theorem somewhere. In this case our embedded manifold is $O(n)$.