  We consider two Riemannian geometries for the manifold $\mathcal{M}(p,m\times
n)$ of all $m\times n$ matrices of rank $p$. The geometries are induced on
$\mathcal{M}(p,m\times n)$ by viewing it as the base manifold of the submersion
$\pi:(M,N)\mapsto MN^T$, selecting an adequate Riemannian metric on the total
space, and turning $\pi$ into a Riemannian submersion. The theory of Riemannian
submersions, an important tool in Riemannian geometry, makes it possible to
obtain expressions for fundamental geometric objects on $\mathcal{M}(p,m\times
n)$ and to formulate the Riemannian Newton methods on $\mathcal{M}(p,m\times
n)$ induced by these two geometries. The Riemannian Newton methods admit a
stronger and more streamlined convergence analysis than the Euclidean
counterpart, and the computational overhead due to the Riemannian geometric
machinery is shown to be mild. Potential applications include low-rank matrix
completion and other low-rank matrix approximation problems.
