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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the singularity of the exponential map on a Lie group


Author: Heng Lung Lai
Journal: Proc. Amer. Math. Soc. 62 (1977), 334-336
MSC: Primary 22E60
DOI: https://doi.org/10.1090/S0002-9939-1977-0432823-4
MathSciNet review: 0432823
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Abstract: Let $ \mathfrak{G}$ be a connected (real or complex) Lie group with Lie algebra G. Define a conjugate point g of $ \mathfrak{G}$ as a point $ g = \exp x$ for some $ x \in G$ and $ d{\exp _x}$ is a noninvertible linear map. We prove that $ g \in \mathfrak{G}$ is a conjugate point if and only if $ g = \exp {x_\lambda }$ for at least a (complex parameter) family of elements $ {x_\lambda }(\lambda \in {\mathbf{C}})$ in G.


References [Enhancements On Off] (What's this?)

  • [1] J. Dixmier, L’application exponentielle dans les groupes de Lie résolubles, Bull. Soc. Math. France 85 (1957), 113–121 (French). MR 0092930
  • [2] SigurÄ‘ur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455

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DOI: https://doi.org/10.1090/S0002-9939-1977-0432823-4
Article copyright: © Copyright 1977 American Mathematical Society