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?
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EDIT, Thursday evening: a good illustration for the bit about negative curvature is the inner circle in the "hole" in a torus of revolution, which is a "closed geodesic" and cannot be contracted continuously to a point. The closed geodesic occurs in the (inner) part of the torus where the Gauss curvature is negative, while the outer portion has positive curvature. Meanwhile, no compact surface in $\mathbb R^3$ can have negative Gauss curvature everywhere, for which Neal gave a simple answer in comments. It is, of course, possible to have noncompact surfaces with negative curvature, such as the catenoid of revolution. However, for any such surface of revolution with negative curvature, the absolute value of the curvature goes to $0$ as we get far from the central axis. So Mariano asked whether there are any infinite surfaces in $\mathbb R^3$ with Gauss curvature that is bounded away from $0,$ and the answer is no, not for a $C^2$ surface. Link to the question MARIANO QUESTION.
ORIGINAL: This is Gauss-Bonnet for polygons on an oriented surface. The familiar case should be this: the area of a geodesic triangle on the unit sphere is $\alpha + \beta + \gamma - \pi,$ where $\alpha, \beta, \gamma$ are the angles at the vertices of the triangle. Next, we define the external angles $\theta_1 = \pi - \alpha, \; \theta_2 = \pi - \beta, \; \theta_3 = \pi - \gamma, $ according to Figure 4-25 of do Carmo:
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The unit sphere has curvature $K=1,$ so the integral of $K$ over a polygon is just its area. Furthermore, the boundary arcs are geodesics, with geodesic curvature $k_g = 0.$ So the statement about this area is equivalent to $$ \int \int_T \; K d \sigma \; + \theta_1 + \theta_2 + \theta_3 = 2 \pi. $$
Compare this with Gauss-Bonnet, formula (1) here:
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You may or may not be told this: in the simply connected hyperbolic plane of constant curvature $-1,$ the area of a triangle is $ \pi - (\alpha + \beta + \gamma),$ where $\alpha, \beta, \gamma$ are the angles at the vertices of the triangle. However, because $K = -1,$ this is once again the same as $$ \int \int_T \; K d \sigma \; + \theta_1 + \theta_2 + \theta_3 = 2 \pi. $$
For your exercise, you are asked about an (oriented) "diangle" $D,$ with angles $0 \leq \alpha, \beta \leq \pi.$ If an angle is equal to $0,$ that is what the author calls a "cusp," see Figure 4-26, the right half. We take the external angles as before, and get $$ \int \int_D \; K d \sigma \; + \theta_1 + \theta_2 = 2 \pi, $$ $$ \int \int_D \; K d \sigma \; + (\pi - \alpha) + (\pi - \beta) = 2 \pi, $$ $$ \int \int_D \; K d \sigma \; = \alpha + \beta. $$ What does this mean? We have $K < 0.$ The right hand side of the equation is nonnegative. If there is any interior to the "diangle," the integral is strictly negative. The only legal possibility is that the two geodesics are identical, both vertices are cusps, the two geodesic arcs are exactly the same, once out, once back.
Otherwise, and here is where the topology comes in, the two geodesic arcs do not bound a piece of surface as I was assuming, and the loop cannot be deformed homotopically to a single point, thus the surface is not simply connected.
Well, I hope that works for you. There is quite a bit of detail that could be added about orientation. Furthermore, I do not think a closed surface in $\mathbb R^3$ can have negative Gauss curvature. I know that a closed surface in $\mathbb R^3$ cannot have constant negative Gauss curvature.