Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Lie groups which admit flat left invariant metrics


Authors: John R. Herring and John J. O’Sullivan
Journal: Proc. Amer. Math. Soc. 82 (1981), 257-260
MSC: Primary 53C20; Secondary 53C30
DOI: https://doi.org/10.1090/S0002-9939-1981-0609662-5
MathSciNet review: 609662
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a Lie group which admits a flat left invariant metric. We show that any nonflat left invariant metric on $ G$ has conjugate points and we describe how some of the conjugate points arise.


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

  • [1] J. Cheeger and D. G. Ebin, Comparison theorems in riemannian property, North-Holland, Amsterdam, 1975.
  • [2] D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75-90. MR 0247590 (40:854)
  • [3] J. R. Herring and J. J. O'Sullivan, The Jacobi equation on Lie groups with left invariant metrics (to appear).
  • [4] E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 47-51. MR 0023591 (9:378d)
  • [5] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. 1, Interscience, New York, 1962.
  • [6] J. Milnor, Curvature of left invariant metrics on Lie groups, Advances in Math. 21 (1976), 293-329. MR 0425012 (54:12970)
  • [7] J. J. O'Sullivan, Manifolds without conjugate points, Math. Ann. 210 (1974), 295-311. MR 0365413 (51:1665)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C20, 53C30

Retrieve articles in all journals with MSC: 53C20, 53C30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0609662-5
Keywords: Lie group, flat left invariant metric, conjugate points
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society