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

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

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]**P. Eberlein,*Geometry of*2-*step nilpotent groups with a left invariant metric*, Ann. Sci. Ecole Norm. Sup. (to appear). MR**1296558 (95m:53059)****[CDKR]**M. Cowling, A. Dooley, A. Koranyi, and F. Ricci,*H-type groups and Iwasawa decompositions*, Adv. Math.**87**(1991), 1-41. MR**1102963 (92e:22017)****[K1]**A. Kaplan,*Riemannian nilmanifolds attached to Clifford modules*, Geom. Dedicata**11**(1981), 127-136. MR**621376 (82h:22008)****[K2]**-,*On the geometry of the groups of Heisenberg type*, Bull. London Math. Soc.**15**(1983), 35-42. MR**686346 (84h:53063)****[Ko]**A. Koranyi,*Geometric properties of Heisenberg type groups*, Adv. Math.**56**(1985), 28-38. MR**782541 (86h:53050)**

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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