This rather theoretical study with practical applications for engineers and physicists is essentially based on both the use of the topological notion of the Euler characteristic and the Gauss-Bonnet theorem to establish various analytical formulas concerning firstly the axisymmetric capillary bridges distortions. In differential geometry, the Gauss-Binet-Bonnet formula connects the integral of the Gauss curvature of a compact two-dimensional surface with boundaries to its topology. It is a suitable and elegant mathematical tool for calculating exactly the value of the bending stress and highlighting its parameters and their respective influence. We establish rigorously in particular the fact that the bending effects measured via the bending stress are proportional to the sum of the cosine values of the two contact angles (possibly distinct, with the same sign or opposite signs depending on the nature of the supports and the affinity between the liquid and the solids). We specify the exact expression of the proportionality constant: a way to structure, on the basis of this quantified criterion, a classification, the right cylinders being then the borderline cases, without bending stress. This returns to the delicate question of the formation of the contact angles, often spontaneous values and a priori unknowns of the problem. The importance of the contact angles is thereby mathematically evident in the writing of the boundary conditions associated to the generalized Young-Laplace equation and in the expression of the bending stress. The method can be generalized beyond the axisymmetric capillary bridges case. For that, one has to associate topological and analytical geometries to calculate the total signed geodesic curvature of the boundaries, not necessarily circular. The value of the contact angles again plays an essential role.