Gauss–Bonnet theorem
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In differential geometry, the Gauss–Bonnet theorem is a fundamental formula which links the curvature of a surface to its underlying topology.
Sculpture made from flat materials using the Gauss–Bonnet Theorem
Geodesic
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In geometry, a geodesic is a curve representing in some sense the locally shortest path between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any di
Gauss–Bonnet theorem
…characteristic is 1. On the left hand side of the theorem, we have K = 1 / R 2 and k g = 0 , because the boundary is the equator and the equator is a geodesic of the sphere. Then, since M has area 4 π R 2 2 , we obtain that ∫ M K d A = 2 π . On the other hand, suppose we flatten the hemisphere to make it into…
Geodesic triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.