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This theorem is baffling. Its Latin wording (“excellent theorem”) was forged by Gauß because he was so excited about it. Gauß studied surfaces for years before discovering this result. It is the kind of theorem which could have waited dozens of years more before being discovered by another mathematician since, unlike so much of intellectual history, it was absolutely not in the air.

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  1. By algebraic measure, we mean (as we discussed in studying turning numbers) that we have to count with opposite signs when we are on regions that get mapped with reversed orientation.

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  2. The function k g is the geodesic curvature introduced on page on page 34.

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  3. Recall that segment is the word for shortest geodesic, that is to say a geodesic with length equal to the distance between its end points.

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  4. Alexandrov proved it even for surfaces which are not very smooth; Cartan had already proven it around 1930 for smooth surfaces with Δ=0

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© 2003 Springer-Verlag Berlin Heidelberg

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Berger, M. (2003). Surfaces from Gauß to Today. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg.

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-65317-2

  • Online ISBN: 978-3-642-18245-7

  • eBook Packages: Springer Book Archive

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