Rectangular matrices

Question

Let $A$ is a $m \times n$ matrix factorized into $A = UV'$ where $U$ is $m \times k$ and $V$ is $n \times k$. I define the group of matrices $(UR,VR^{-1})$ where $R \in O(k)$ i.e. orthogonal group of $k \times k$. Since for any value of $R$, the factorization holds true, can we say that this group forms a manifold ?. If so, what will be dimension of this manifold ?