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Präsentation des Textes

  • 2025
  • OriginalPaper
  • Chapter
Published in:
Author
Jürgen Jost
Published in
Bernhard Riemann - Über die Hypothesen, welche der Geometrie zu Grunde liegen
Publisher
Springer Berlin Heidelberg

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Abstract

Dieses Kapitel erläutert sorgfältig den Riemannschen Text. Nach einer verbalen Beschreibung werden die mathematischen Strukturen, die hinter Riemanns Überlegungen standen oder von seinen Nachfolgern entwickelt wurden, detailliert dargestellt. Zentrale Begriffe sind die (differenzierbare) Mannigfaltigkeit, der metrische Tensor, geodätische Linien und vor allem der Riemannsche Krümmungstensor. Die aus diesem Tensor gewonnenen Schnittkrümmungen bilden einen vollständigen Satz von lokalen Invarianten, also koordinatenunabhängigen Größen, für eine Riemannsche Metrik. Dies ist eine zentrale Erkenntnis von Riemann. In diesem Kapitel wird daher der mathematische Formalismus vollständig entwickelt.

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previous chapter Riemanns Habilitationsvortrag
next chapter Rezeptions- und Wirkungsgeschichte
Title
Präsentation des Textes
Author
Jürgen Jost
Copyright Year
2025
Publisher
Springer Berlin Heidelberg
Book
Bernhard Riemann - Über die Hypothesen, welche der Geometrie zu Grunde liegen

Print ISBN: 978-3-662-70871-2

Electronic ISBN: 978-3-662-70872-9

Copyright Year: 2025

DOI
https://doi.org/10.1007/978-3-662-70872-9_4

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