The goal of this article is to study the space of smooth Riemannian structures on compact manifolds with boundary that satisfies a critical point equation associated with a boundary value problem. We provide an integral formula which enables us to show that if a critical metric of the volume functional on a connected n n -dimensional manifold M n M^n with boundary ∂ M \partial M has parallel Ricci tensor, then M n M^n is isometric to a geodesic ball in a simply connected space form R n \mathbb {R}^{n} , H n \mathbb {H}^{n} or S n \mathbb {S}^{n} .