Lie groups which admit flat left invariant metrics

Herring, John R.; O’Sullivan, John J.

doi:10.1090/S0002-9939-1981-0609662-5

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

[1] J. Cheeger and D. G. Ebin, Comparison theorems in riemannian property, North-Holland, Amsterdam, 1975.
[2] Detlef Gromoll and Wolfgang Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75–90. MR 0247590, https://doi.org/10.2307/1970682
[3] J. R. Herring and J. J. O'Sullivan, The Jacobi equation on Lie groups with left invariant metrics (to appear).
[4] Eberhard Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U. S. A. 34 (1948), 47–51. MR 0023591
[5] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. 1, Interscience, New York, 1962.
[6] John Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293–329. MR 0425012, https://doi.org/10.1016/S0001-8708(76)80002-3
[7] John J. O’Sullivan, Manifolds without conjugate points, Math. Ann. 210 (1974), 295–311. MR 0365413, https://doi.org/10.1007/BF01434284