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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Connections on semisimple Lie groups


Author: Robert E. Beck
Journal: Trans. Amer. Math. Soc. 164 (1972), 453-460
MSC: Primary 22.80
MathSciNet review: 0289727
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Abstract: The plus and minus connections of Cartan and Schouten, which exist on any Lie group, have the following three properties: (1) the connection is left invariant, (2) the curvature of the connection is zero, (3) the set of maximal geodesics through the identity of the Lie group is equal to the set of one-parameter subgroups of the Lie group. It is shown that the plus and minus connections are the only ones with these properties on a real simple Lie group. On a real semisimple Lie group the connections with these properties are in one-to-one correspondence with the ways of choosing an ideal of the Lie algebra and then choosing a complementary subspace to it.


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

  • [1] E. Cartan and J. Schouten, On the geometry of the group-manifold of simple and semisimple groups, Nederl. Akad. Wetensch. Proc. Ser. A 29 (1926), 803-815.
  • [2] Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
  • [3] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
  • [4] Katsumi Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33–65. MR 0059050

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0289727-X
Keywords: Real semisimple Lie group, affine connection, real semisimple Lie algebra, skew-symmetric representation
Article copyright: © Copyright 1972 American Mathematical Society