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Minimal foliations on Lie groups. (English) Zbl 0766.53021

Let \(G\) be a connected Lie group, \(H\) a connected, not necessarily closed, Lie subgroup of \(G\). The authors discuss minimality of foliations \(F(G,H)\). In R. Takagi and S. Yorozu [Tohoku Math. J., II. Ser. 36, 541-554 (1984; Zbl 0559.53018)] it is proved that if a left invariant Riemannian metric \(g\) is \(\text{Ad}(H)\)-invariant then \(g\) is bundle-like for \(F(G,H)\) and \(F(G,H)\) is totally geodesic for \(g\). In this paper, the authors prove that there always exists a (bundle-like) Riemannian metric on \(G\) for which the leaves of \(F(G,H)\) are minimal submanifolds.

MSC:

53C12 Foliations (differential geometric aspects)
57R30 Foliations in differential topology; geometric theory
22E15 General properties and structure of real Lie groups

Citations:

Zbl 0559.53018
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References:

[1] Fédida, E., Feuilletages de Lie (1983), Thèse Strasbourg · Zbl 0218.57014
[2] Ghys, E., Feuilletages riemanniens sur les variétés simplement connexes, Ann. Inst. Fourier, Grenoble, 34, 4, 203-223 (1984) · Zbl 0525.57024
[3] Haefliger, A., Some remarks on foliations with minimal leaves, Journal of Diff. Geom., 15, 269-284 (1980) · Zbl 0444.57016
[4] Maľcev, A., On the simple connectedness of invariant subgroups of the Lie groups, Acad. Sci. U.R.S.S., 34, 10-13 (1942) · Zbl 0061.04605
[5] Molino, P., Riemannian Foliations, Progress in Math., Vol. 73 (1988), Birkaüser
[6] Takagi, R.; Yorozu, S., Minimal foliations on Lie groups, Tohoku Math. J., 36, 541-554 (1984) · Zbl 0559.53018
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