Special properties of realizations of supersymmetry on noncompact manifolds are discussed. On the basis of the supersymrnetric scattering theory and the supersymmetric trace formulas, the absolute or relative Euler characteristic of a barrier in R^N can be obtained from the scattering data for the Laplace operator on forms with absolute or relative boundary conditions. An analog of the Chern-Gauss-Bonnet theorem for noncompact manifolds is also obtained. The map from the stationary curve of an antiholomorphic involution on a compact Riemann surface to the real circle on the Riemann sphere, generated by a real meromorphic function is considered. An analytic expression for its topological index is obtained by using supersymmetric quantum mechanics with meromorphic superpotential on the klein surface.