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Riemann’s Blueprints for Architecture in Myriad Dimensions

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A Panoramic View of Riemannian Geometry
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Abstract

As we said in chapter 2, Riemann’s construction of the Riemannian manifold consisted first in building the foundation of the smooth manifold. He then established on that foundation the concept of a Riemannian metric. In the first two sections we will present smooth manifolds, and thereafter define Riemannian metrics. The notion of smooth manifold is at the same time extremely natural and quite hard to define correctly. This notion started with Riemann in 1854 and was widely used. Hermann Weyl was the first to lay down solid foundations for this notion in 1923. The definition became completely clear in the famous article Whitney 1936 [1259].

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References

  1. See page 150 for the definition of simple connectivity.

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  2. The group of diffeomorphisms of S 2, or of any other manifold, is a very large group.

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  3. The coordinates are, for y ∈ En +1 with y = χ: \( x = \left( {\frac{{y1}}{{\rho - y_{n + 1} }},...,\frac{{y1}}{{\rho - y_{n + 1} }}} \right). \)

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  4. and even a fifth model in §§4.3.4

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  5. See Spivak 1979 [1155], volume 2, for a guess as to how Riemann achieved this—he did not give any detailed computation in his text.

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  6. This work of Samelson was the first appearance of the concept of Riemannian submersion (see §§4.3.6).

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© 2003 Springer-Verlag Berlin Heidelberg

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Berger, M. (2003). Riemann’s Blueprints for Architecture in Myriad Dimensions. In: A Panoramic View of Riemannian Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18245-7_4

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  • DOI: https://doi.org/10.1007/978-3-642-18245-7_4

  • 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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