continuous scalar function. In this approach, a series of isosurfaces are generated and systematically studied for eigenvalue and curvature maps. Geometric and topological (Geo-Topo) fingerprints are identified to characterize unique patterns within eigenvalue and curvature maps, specifically, the maps of three eigenvalues derived from local Hessian matrix at each location of the rigidity density and the maps of Gaussian, mean, maximal and minimal curvatures computed everywhere of the rigidity density. Topological properties of eigenvalue and curvature maps are classified by their critical points. The evolution of isosurfaces during the filtration process is found to be well characterized by CPs. Different behaviors are found in different types of maps. Persistent homology analysis is also employed for eigenvalue map analysis to reveal intrinsic topological invariants of three Hessian matrix eigenvalues. Finally, a new minimal molecular surface, called analytical minimal molecular surface (AMMS) via the zero-value isosurface of the mean curvature map, has been introduced. It is found that this new surface definition can capture the topological property, such as the inner bond information. It also offers an efficient geometric modeling of biomolecules.

The rest of this paper is organized as following: Section \[sec:discrete\] is devoted to a review of some discrete mathematical 