Obata-Type Rigidity on Static Manifolds with Boundary

We investigate static metrics on simple manifolds with compact boundary and establish an Obata-type rigidity theorem. We identify new sufficient geometric conditions under which the combined curvature map $g\mapsto (R_g, H_g)$ is a local surjection. Consequently, we demonstrate that in contrast to manifolds without boundary, where staticity obstructs deformability, the scalar curvature map can be locally surjective at static metrics on manifolds with boundary.