Geometric and electrostatic modeling is an essential component in computational biophysics and molecular biology. Commonly used geometric representations admit geometric singularities such as cusps, tips and self-intersecting facets that lead to computational instabilities in the molecular modeling. The present work explores the use of flexibility and rigidity index (FRI), which has a proved superiority in protein B-factor prediction, for biomolecular geometric representation and associated electrostatic analysis. FRI rigidity surfaces are free of geometric singularities. We proposed a rigidity based Poisson -Boltzmann equation for biomolecular electrostatic analysis. Our approaches to surface and electrostatic modeling are validated by a set of 21 proteins. Our results are compared with those of established methods. Finally, being smooth and analytically differentiable, FRI rigidity functions offer excellent curvature analysis, which characterizes concave and convex regions on protein surfaces. Polarized curvatures constructed by using the product of minimum curvature and electrostatic potential is shown to predict potential protein-ligand binding sites. (C) 2016 Elsevier B.V. All rights reserved.