<p>Static manifolds with boundary appear naturally in the context of the prescribed scalar curvature problem on manifolds with boundary, when the mean curvature of the boundary is also prescribed. They also arise in the setting of general relativity: For example, the time slice of the photon sphere on the Riemannian Schwarzschild manifold splits it into static manifolds with boundary. In this paper, we prove a number of theorems that relate the topology and geometry of a given static manifold with boundary to some properties of the zero-level set of its potential (such as connectedness and closedness). Also, we characterize the round ball in Euclidean 3-space with standard potential as the only scalar-flat static manifold with mean-convex boundary whose zero-level set of the potential has Morse index one. This result follows from a general isoperimetric inequality for 3-dimensional static manifolds with boundary, whose zero-level set of the potential has Morse index one. Finally, we prove some uniqueness theorems for the domains bounded by the photon sphere on the Riemannian Schwarzschild manifold.</p>