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Transactions of the American Mathematical Society

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



Geometrizing Infinite Dimensional Locally Compact Groups

Author: Conrad Plaut
Journal: Trans. Amer. Math. Soc. 348 (1996), 941-962
MSC (1991): Primary 53C70, 22D05; Secondary 22E65
MathSciNet review: 1348156
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Abstract: We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.

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  • 1. Alexandrov, A.D., Die innere Geometrie der konvexen Flächen, Akad. Verl., Berlin, 1955. MR 17:74
  • 2. Alexandrov, A.D., Berestovskii, V.N., and Nikolaev, I.G., Generalized Riemannian spaces, Russian Math Surveys 41 (1986) 1-54. MR 88e:53103
  • 3. Berestovskii, V. N., Spaces with bounded curvature and distance geometry, Siberian Math. J. 27 (1986) 8-19. MR 89k:53063
  • 4. Berestovskii, V. N., Homogeneous spaces with intrinsic metric, Soviet Math. Dokl. 27 (1989) 60-63.
  • 5. Berestovskii, V. N., and Plaut, C., Homogeneous spaces of curvature bounded below, preprint.
  • 6. Cheeger, J., and Ebin, D., Comparison theorems in Riemannian geometry, North-Holland Math Library 9 (1975), Amsterdam. MR 56:16538
  • 7. Burago, Yu., Gromov, M., and Perelman, G., A.D. Alexandrov spaces with curvature bounded below, Russian Math Surveys 47 (1992) 1-58. MR 93m:53035
  • 8. Cohn-Vossen, S., Existenz kürzester Wege, Dokl. Math. SSSR 8 (1935) 339-342.
  • 9. Gluskov, V.N., The structure of locally compact groups and Hilbert's Fifth Problem, AMS Translations 15 (1960) 55-94. MR 22:5690
  • 10. Gromov, M., Lafontaine, J., and Pansu, P., Structures métrique pours les variétés riemanniannes, Textes Math., vol. 1 CEDIC/Fernand Nathan, 1981. MR 85e:53051
  • 11. Grove, K., Metric differential geometry, Differential Geometry, Lecture Notes in Math., vol. 1263, Springer (1987) 171-227. MR 88i:53075
  • 12. Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978. MR 80k:53081
  • 13. Hurewicz, W. and Wallman, H., Dimension Theory, Princeton U. Press, Princeton, 1941. MR 3:312b
  • 14. Lashof, R., Lie algebras of locally compact groups, Pac. J. Math. 7 (1957) 1145-1162. MR 19:1064a
  • 15. Montgomery, D. and Zippin, L., Topological transformation groups, Interscience, New York, 1955. MR 17:383b
  • 16. O'Neill, B., The fundamental equations of a sumersion, Michigan Math J. 13 (1966), 459-469. MR 34:751
  • 17. Plaut, C., Almost Riemannian spaces, J. Diff. Geom. 34 (1991) 515-537. MR 92h:53047
  • 18. Plaut, C., Metric curvature, convergence, and topological finiteness, Duke Math. J. 66 (1992) 43-57. MR 93e:53051
  • 19. Plaut, C., Metric pinching of locally symmetric spaces, Duke Math. J. 73 (1994) 155-162. MR 95c:53055
  • 20. Plaut, C., Correction to ``Metric pinching of locally symmetric spaces'', Duke Math. J., 75 (1994) 527-528. CMP 94:17
  • 21. Plaut, C., Spaces of Wald curvature bounded below, J. Geom. Analysis, to appear.
  • 22. Rickert, N. W., Arcs in Locally Compact Groups, Math. Annalen 172 (1967), 222-228. MR 35:4331
  • 23. Rickert, N., Some Properties of Locally Compact Groups , J. Austral. Math. Soc. 7 (1967) 433-454. MR 36:2735
  • 24. Rinow, W., Die Innere Geometrie der Metrischen Räume, Springer, Berlin, 1961. MR 23:A1290
  • 25. van Kampen, E. R., The structure of a compact connected group, Amer. J. Math. 57 (1935) 301-308.

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Additional Information

Conrad Plaut
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300

Keywords: Locally compact groups, Alexandrov curvature, invariant metric
Received by editor(s): February 16, 1994
Additional Notes: The author gratefully acknowledges the support of NSF grant DMS-9401302
Article copyright: © Copyright 1996 American Mathematical Society