It is well known that the constant mean curvature surfaces, highly studied, are obtained by minimizing the only surface tension energy at fixed volume, the constant corresponding to the Lagrange multiplier. Implicitly, this means that the Gaussian curvature and the total curvature are not taken into account and that therefore, the bending energy is disregarded or a priori considered as having negligible effects compared to the effects of surface tension. Admittedly, the spherical or distorted water drops, freely evolving in the air, exactly agree with this simplifying method. Indeed, under boundaryless manifold condition, the most common formulation of the Gauss-Bonnet integration theorem indicates that for a spherical drop or a soap bubble without contact, with or without bump, the integral of the Gaussian curvature over the surface, proportional to the bending energy, is invariant if one bends and deforms the surface (this value is a topological invariant). However, this result is not valid when the drop is subjected to contact boundary conditions. Therefore, because of the constrained boundaries, the study of capillary bridges requires to take into account the Gaussian curvature of the surface and the total geodesic curvature of the boundaries. We study here, theoretically, the real effects of bending forces and therefore of the Gaussian curvature in order to identify a discriminating criterion according to the sign of the Gaussian curvature. Concerning the bending effects, the true shape of the static bridge surfaces is here described by parametric equations, generalizing Delaunay formulas. The related generalized Young-Laplace boundary value system is then solved as an inverse problem from experimental data for the unknown parameters identification. The subject of this study presents strong correspondences with the Gullstrand equation of geometrical optics, with also the gravitational bending angle of light for finite distance and in geometrical approachs to gravitational lensing theory in the astrophysical context.