The relationship between the surface geometry and certain electronic properties is considered for neutral metal systems. For systems enclosed by surfaces of constant curvature, the total energy, the surface energy, and the chemical potential are found to depend linearly on the surface curvature. Explicit expressions are found for the coefficients of this dependence, in particular for the curvature energy. It is shown that for systems with surfaces of varying curvature a surface charge distribution is formed (and therefore an electric potential varying across the surface), to ensure a constant chemical potential. Implications for the ionization potentials of finite and infinite systems of finite curvature (e.g., spheroids and thin wires) are discussed.