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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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
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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