The Gauss-Bonnet Theorem

  • John M. Lee
Part of the Graduate Texts in Mathematics book series (GTM, volume 176)


We are finally in a position to prove our first major local-global theorem in Riemannian geometry: the Gauss-Bonnet theorem. This is a local-global theorem par excellence, because it asserts the equality of two very differently defined quantities on a compact, orientable Riemannian 2-manifold M: the integral of the Gaussian curvature, which is determined by the local geometry of M; and times the Euler characteristic of M, which is a global topological invariant. Although it applies only in two dimensions, it has provided a model and an inspiration for innumerable local-global results in higher-dimensional geometry, some of which we will prove in Chapter 11.


Euler Characteristic Plane Geometry Interior Angle Angle Function Geodesic Triangle 
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Copyright information

© Springer-Verlag New York, Inc. 1997

Authors and Affiliations

  • John M. Lee
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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