Abstract
This paper proves existence and uniqueness of geodesics through singular points on a real analytic surface M with a real analytic symmetric two-tensor g when the tangent space to the singular set is isotropic at the singular point. This is achieved by introducing a new differentiable structure at the singular point and by introducing the concept semi-analytic curves. In this new differentiable structure the geodesic through the noncritical singular point is up to reparameterization of a semi-analytic curve. This makes it possible to prove existence and uniqueness of geodesics through the singular point. The setting in higher dimensions is indicated. Using the same techniques we prove the existence of three collision orbits for the Hamiltonian vector field representing Coulomb forces of the Helium atom.
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