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Der Riemannsche Raum

In: Die Welt der Vektoren

Author

Listed:
  • Franz Ollendorff

    (The Hebrew Technical College, Elektrotechnischen Laboratoriums)

Abstract

Zusammenfassung In einem dreidimensionalen Raume (R3) orientieren wir uns mittels eines Kartesischen Koordinatensystemes xi (i = 1, 2, 3). Zur Ausführung von Messungen bedienen wir uns eines starren Einheits-Maßstabes. Mit seiner Hilfe ermittein wir die (skalare) Entfernung ds zweier infinitesimal benachbarter Punkte. Wir suchen sie mit den am Koordinaten-Gerüst „eingravierten“ Differenzen dxi ihrer Lagekoordinaten in Beziehung zu setzen, deren „Eichung“ mit ebendemselben Einheits-Maßstab vorgenommen wurde. Finden wir hierbei stets (VII 1, 1) $$d{s^2} = \sum\limits_i {d{x^i}d{x^i}} $$ unabhängig von der Lage des Maßstabes relativ zum Ursprung des Bezugs-systemes wie auch von seiner Orientierung relativ zu den Achsen, so ist hierdurch die Geometrie des untersuchten R3 als Euklid ische definiert.

Suggested Citation

  • Franz Ollendorff, 1950. "Der Riemannsche Raum," Springer Books, in: Die Welt der Vektoren, chapter 0, pages 315-374, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7091-7755-6_8
    DOI: 10.1007/978-3-7091-7755-6_8
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