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Published in:
2014 | OriginalPaper | Chapter
7. Elements of the Global Theory of Surfaces
Author : Francis Borceux
Published in: A Differential Approach to Geometry
Publisher: Springer International Publishing
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Abstract
Global theory of surfaces is interested in those properties which refer to wide pieces of the surface, not just to the neighborhood of each point. We study surfaces of revolution, ruled surfaces, developable surfaces. We study when two surfaces are just an “isometric deformation” of each other and establish the classification of developable surfaces. We pay special attention to the surfaces with constant Gaussian curvature and prove the Liebmann characterization of the sphere. We conclude with the study of polygonal decompositions, the Gauss–Bonnet theorem and the Euler–Poincaré characteristic.