A relation between two biharmonic Green’s functions on Riemannian manifolds

The biharmonic Green’s function $\gamma$ whose values and Laplacian are identically zero on the boundary of a region and the biharmonic Green’s function $\Gamma$ whose values and normal derivative vanish on the boundary originated in the investigation of thin plates whose edges are simply supported or clamped, respectively. A relation between these two biharmonic Green’s functions known for planar regions is extended to Riemannian manifolds thereby establishing that any Riemannian manifold for which $\gamma$ exists must also carry $\Gamma$.