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A note on semisimple flat homogeneous spaces

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1045)

Abstract

Let Lo be the isotropy group of the origin in a semisimple flat homogeneous space M=L/Lo associated with a semisimple graded transitive Lie algebra l = g−1⊕g0⊕g1. It is a well known fact, due to T. Ochiai, that Lo can be considered as a Lie subgroup of G2(m), the Lie group of 2-jets j 20 (g) at 0 ∈ g−1, where g is a diffeomorphism from a neighbourhood of 0 ∈ g−1R m onto a neighbourhood of 0 ∈g−1. The aim of this note is to show that Lo is in fact a closed subgroup when it has only a finite number of connected components, giving an affirmative answer to a conjecture suggested to us by M. Takeuchi one year ago.

Keywords

  • Homogeneous Space
  • Closed Subgroup
  • Projective Connection
  • Linear Isotropy
  • Strong Deformation Retract

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© 1984 Springer-Verlag

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Lopera, J.F.T. (1984). A note on semisimple flat homogeneous spaces. In: Naveira, A.M. (eds) Differential Geometry. Lecture Notes in Mathematics, vol 1045. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072171

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  • DOI: https://doi.org/10.1007/BFb0072171

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-12882-3

  • Online ISBN: 978-3-540-38766-4

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