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Lie Groups pp 109-121 | Cite as

Geodesics and Maximal Tori

  • Daniel Bump
Chapter
Part of the Graduate Texts in Mathematics book series (GTM, volume 225)

Abstract

An important theorem of Cartan asserts that any two maximal tori in a compact Lie group are conjugate. There are different ways of proving this. We will deduce it from the surjectivity of the exponential map, which we will prove by showing that a geodesic between the origin and an arbitrary point of the group has the form \(t\mapsto {\mathrm{e}}^{tX}\) for some X in the Lie algebra.

Keywords

Riemannian Manifold Tangent Space Tangent Vector Short Length Unit Normal Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 110.
    S. Kobayashi and K. Nomizu. Foundations of Differential Geometry. Vol I. Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Daniel Bump
    • 1
  1. 1.Department of MathematicsStanford UniversityStanfordUSA

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