Geometry of $2$-step nilpotent groups with a left invariant metric. II
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- by Patrick Eberlein
- Trans. Amer. Math. Soc. 343 (1994), 805-828
- DOI: https://doi.org/10.1090/S0002-9947-1994-1250818-2
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Abstract:
We obtain a partial description of the totally geodesic submanifolds of a 2-step, simply connected nilpotent Lie group with a left invariant metric. We consider only the case that N is nonsingular; that is, ${\text {ad}}\xi :\mathcal {N} \to \mathcal {Z}$ is surjective for all elements $\xi \in \mathcal {N} - \mathcal {Z}$, where $\mathcal {N}$ denotes the Lie algebra of N and $\mathcal {Z}$ denotes the center of $\mathcal {N}$. Among other results we show that if H is a totally geodesic submanifold of N with $\dim H \geq 1 + \dim \mathcal {Z}$, then H is an open subset of $g{N^\ast }$, where g is an element of H and ${N^\ast }$ is a totally geodesic subgroup of N. We find simple and useful criteria that are necessary and sufficient for a subalgebra ${\mathcal {N}^\ast }$ of $\mathcal {N}$ to be the Lie algebra of a totally geodesic subgroup ${N^\ast }$. We define and study the properties of a Gauss map of a totally geodesic submanifold H of N. We conclude with a characterization of 2-step nilpotent Lie groups N of Heisenberg type in terms of the abundance of totally geodesic submanifolds of N.References
- Patrick Eberlein, Geometry of $2$-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 5, 611–660. MR 1296558
- Michael Cowling, Anthony H. Dooley, Adam Korányi, and Fulvio Ricci, $H$-type groups and Iwasawa decompositions, Adv. Math. 87 (1991), no. 1, 1–41. MR 1102963, DOI 10.1016/0001-8708(91)90060-K
- Aroldo Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), no. 2, 127–136. MR 621376, DOI 10.1007/BF00147615
- Aroldo Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), no. 1, 35–42. MR 686346, DOI 10.1112/blms/15.1.35
- Adam Korányi, Geometric properties of Heisenberg-type groups, Adv. in Math. 56 (1985), no. 1, 28–38. MR 782541, DOI 10.1016/0001-8708(85)90083-0
Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 343 (1994), 805-828
- MSC: Primary 53C30; Secondary 22E25
- DOI: https://doi.org/10.1090/S0002-9947-1994-1250818-2
- MathSciNet review: 1250818