A metric is introduced on the two–dimensional space of parameters describing the Ising model on a Bethe lattice of co–ordination number q. The geometry associated with this metric is analysed and it is shown that the Gaussian curvature diverges at the critical point. For the special case q = 2 the curvature reduces to an already known result for the one–dimensional Ising model. The Gaussian curvature is also calculated for a general ferromagnet near its critical point, generalizing a previous result for T > Tc. The general expression near a critical point is compared with the specific case of the Bethe lattice and a subtlety, associated with the fact that the specific heat exponent for the Bethe lattice vanishes, is resolved.