Lie Groups pp 109-121 | Cite as

Geodesics and Maximal Tori

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


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.


Riemannian Manifold Tangent Space Tangent Vector Short Length Unit Normal Vector 
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  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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