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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Geometry of $ 2$-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, $ {\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.


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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