Geometry of -step nilpotent groups with a left invariant metric. II

Author:
Patrick Eberlein

Journal:
Trans. Amer. Math. Soc. **343** (1994), 805-828

MSC:
Primary 53C30; Secondary 22E25

MathSciNet review:
1250818

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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, is surjective for all elements , where denotes the Lie algebra of *N* and denotes the center of . Among other results we show that if *H* is a totally geodesic submanifold of *N* with , then *H* is an open subset of , where *g* is an element of *H* and is a totally geodesic subgroup of *N*. We find simple and useful criteria that are necessary and sufficient for a subalgebra of to be the Lie algebra of a totally geodesic subgroup . 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*.

**[E]**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****[CDKR]**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**, 10.1016/0001-8708(91)90060-K**[K1]**Aroldo Kaplan,*Riemannian nilmanifolds attached to Clifford modules*, Geom. Dedicata**11**(1981), no. 2, 127–136. MR**621376**, 10.1007/BF00147615**[K2]**Aroldo Kaplan,*On the geometry of groups of Heisenberg type*, Bull. London Math. Soc.**15**(1983), no. 1, 35–42. MR**686346**, 10.1112/blms/15.1.35**[Ko]**Adam Korányi,*Geometric properties of Heisenberg-type groups*, Adv. in Math.**56**(1985), no. 1, 28–38. MR**782541**, 10.1016/0001-8708(85)90083-0

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1250818-2

Keywords:
2-step nilpotent Lie group,
left invariant metric,
totally geodesic subgroup,
totally geodesic submanifold,
Gauss map,
Heisenberg type

Article copyright:
© Copyright 1994
American Mathematical Society