What ideas/formulas are required to solve this? Exercise: If a and b are two geodesics from point p to q, how do you prove that M is not simply connected? M is a regular surface in R3 and has negative Gaussian curvature K. Points p & q are two distinct points on surface M.
I know surface is simply connected if any simple closed curve can be shrunk to a point continuously in the set. So a sphere is simply connected, and a donut is not simply connected. And I know that since the curvature is negative, and since K=det(Sp)=w*z, so eigenvalues w and z have opposite signs. I also have the formula K=(eg-f^2)/(EG-F^2) based on the 1st fundamental forms and derivatives for solving for E,F,G,e,f,g. Not sure what if anything that is used to solve. And the only geodesic equations I can think of off hand are
d/dt(E u' + F v')=1/2[E_u u'^2 + 2F_u u'v' +G_u v'^2 ], d/dt(Fu'+Gv')=1/2[E_v u'^2 +2F_u u'v' +G_v v'^2 ], and geodesic curvature c=, and Gauss-Bonnet integrals but not sure how that can be applied. If they can be applied, please let me know. Maybe the fact that the normal of any point on the geodesic arc also is on the same normal vector to the surface M at that point helps? What ideas and equations are required to solve the exercise?