Our goal for the remainder of this book is to generalize to higher dimensions some of the geometric and topological consequences of the Gauss-Bonnet theorem. We need to develop a new approach: instead of using Stokes’s theorem and differential forms to relate the curvature to global topology as in the proof of the Gauss-Bonnet theorem, we study how curvature affects the behavior of nearby geodesies. Roughly speaking, positive curvature causes nearby geodesies to converge (Figure 10.1), while negative curvature causes them to spread out (Figure 10.2). In order to draw topological consequences from this fact, we need a quantitative way to measure the effect of curvature on a one-parameter family of geodesies.
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