Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-06T07:46:32.138Z Has data issue: false hasContentIssue false
  • English
  • Français

Some embeddings of Lie groups in Euclidean space

Part of: Lie groups

Published online by Cambridge University Press:  26 February 2010

Elmer Rees
Elmer Rees
Affiliation:
University College of Swansea Wales.
Get access

Extract

Every n-dimensional manifold admits an embedding in R2n by the result of H. Whitney [11]. Lie groups are parallelizable and so by the theorem of M. W. Hirsch [5] there is an immersion of any Lie group in codimension one. However no general theorem is known which asserts that a parallelizable manifold embeds in Euclidean space of dimension less than 2n. Here we give a method for constructing smooth embeddings of compact Lie groups in Euclidean space. The construction is a fairly direct one using the geometry of the Lie group, and works very well in some cases. It does not give reasonable results for the group Spin (n) except for low values of n. We also give a method for constructing some embeddings of Spin (n), this uses the embedding of SO(n) that was constructed by the general method and an embedding theorem of A. Haefliger [3]. Although this is a very ad hoc method, it has some interest as it seems to be the first application of Haefliger's theorem which gives embedding results appreciably below twice the dimension of the manifold. The motivation for this work was to throw some light on the problem of the existence of low codimensional embeddings of parallelizable manifolds.

MSC classification

Secondary: 22E99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 18 , Issue 1 , June 1971 , pp. 152 - 156
Copyright
Copyright © University College London 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Borel, A. and Hirzebruch, F., “Characteristic classes and homogeneous spaces, III”, Amer. J. Math., 82 (1960), 491–504.CrossRefGoogle Scholar
2.Dieck, T. torn, “Klassifikation numerierbar Bündel”, Arch. Math. (Basel), 17 (1966), 395–399.CrossRefGoogle Scholar
3.Haefliger, A., “Plongements differentiables dans le domaine stable”, Comm. Math. Helv., 37 (1962/1963), 155–176.CrossRefGoogle Scholar
4.Hantzsche, W., “Einlagerung von Mannigfaltikeiten in euklidische Raume”, Math. Z., 43 (1938), 38–58.CrossRefGoogle Scholar
5.Hirsch, M. W., “Immersions of manifolds”, Trans. Amer. Math. Soc, 93 (1959), 242–279.CrossRefGoogle Scholar
6.Husemoller, D., Fibre Bundles (McGraw-Hill, 1966).CrossRefGoogle Scholar
7.Mahowald, M., “On embedding manifolds which are bundles over spheres”, Proc. Amer. Math. Soc, 15 (1964), 579–583.CrossRefGoogle Scholar
8.Milnor, J., “Construction of universal bundles, II”, Ann. of Math. (2), 63 (1956), 430–436.CrossRefGoogle Scholar
9.Tits, J., Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen. Lecture Notes in Mathematics, vol. 40 (Springer-Verlag, 1967).CrossRefGoogle Scholar
10.Tornehave, J., “Immersions of complex flag manifolds “, Math. Scand., 23 (1968), 22–26.CrossRefGoogle Scholar
11.Whitney, H., “The self-intersections of a smooth n-manifold in 2n-space”, Ann. of Math. (2), 45 (1944), 220–246.CrossRefGoogle Scholar