Geometrizing Infinite Dimensional Locally Compact Groups
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- by Conrad Plaut
- Trans. Amer. Math. Soc. 348 (1996), 941-962
- DOI: https://doi.org/10.1090/S0002-9947-96-01592-9
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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.References
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- A. D. Aleksandrov, V. N. Berestovskiĭ, and I. G. Nikolaev, Generalized Riemannian spaces, Uspekhi Mat. Nauk 41 (1986), no. 3(249), 3–44, 240 (Russian). MR 854238
- V. N. Berestovskiĭ, Homogeneous spaces with an intrinsic metric, Dokl. Akad. Nauk SSSR 301 (1988), no. 2, 268–271 (Russian); English transl., Soviet Math. Dokl. 38 (1989), no. 1, 60–63. MR 967817
- Berestovskii, V. N., Homogeneous spaces with intrinsic metric, Soviet Math. Dokl. 27 (1989) 60-63.
- Berestovskii, V. N., and Plaut, C., Homogeneous spaces of curvature bounded below, preprint.
- Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
- Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
- Cohn-Vossen, S., Existenz kürzester Wege, Dokl. Math. SSSR 8 (1935) 339-342.
- V. M. Gluškov, The structure of locally compact groups and Hilbert’s fifth problem. , Amer. Math. Soc. Transl. (2) 15 (1960), 55–93. MR 0114872, DOI 10.1090/trans2/015/04
- Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
- Karsten Grove, Metric differential geometry, Differential geometry (Lyngby, 1985) Lecture Notes in Math., vol. 1263, Springer, Berlin, 1987, pp. 171–227. MR 905882, DOI 10.1007/BFb0078613
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
- Conrad Plaut, Almost Riemannian spaces, J. Differential Geom. 34 (1991), no. 2, 515–537. MR 1131442
- Conrad Plaut, Metric curvature, convergence, and topological finiteness, Duke Math. J. 66 (1992), no. 1, 43–57. MR 1159431, DOI 10.1215/S0012-7094-92-06602-6
- Conrad Plaut, Metric pinching of locally symmetric spaces, Duke Math. J. 73 (1994), no. 1, 155–162. MR 1257280, DOI 10.1215/S0012-7094-94-07305-5
- Plaut, C., Correction to “Metric pinching of locally symmetric spaces”, Duke Math. J., 75 (1994) 527-528.
- Plaut, C., Spaces of Wald curvature bounded below, J. Geom. Analysis, to appear.
- Neil W. Rickert, Arcs in locally compact groups, Math. Ann. 172 (1967), 222–228. MR 213467, DOI 10.1007/BF01351189
- Neil W. Rickert, Some properties of locally compact groups, J. Austral. Math. Soc. 7 (1967), 433–454. MR 0219656, DOI 10.1017/S1446788700004389
- Willi Rinow, Die innere Geometrie der metrischen Räume, Die Grundlehren der mathematischen Wissenschaften, Band 105, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961. MR 0123969, DOI 10.1007/978-3-662-11499-5
- van Kampen, E. R., The structure of a compact connected group, Amer. J. Math. 57 (1935) 301-308.
Bibliographic Information
- Conrad Plaut
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
- Email: plaut@novell.math.utk.edu
- Received by editor(s): February 16, 1994
- Additional Notes: The author gratefully acknowledges the support of NSF grant DMS-9401302
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 941-962
- MSC (1991): Primary 53C70, 22D05; Secondary 22E65
- DOI: https://doi.org/10.1090/S0002-9947-96-01592-9
- MathSciNet review: 1348156