Rocky Mountain Journal of Mathematics

Totally geodesic subalgebras in 2-step nilpotent Lie algebras

Rachelle C. Decoste and Lisa Demeyer

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Abstract

We describe totally geodesic subalgebras of a metric 2-step nilpotent Lie algebra $\n$. We prove that a totally geodesic subalgebra of $\n$ is either abelian and flat or can be decomposed as a direct sum determined by the curvature transformation. In addition, we give conditions under which a totally geodesic submanifold of a simply connected 2-step nilpotent Lie group is a totally geodesic subgroup. We follow Eberlein's 1994 paper in which he imposes the condition of nonsingularity on $\n$. We remove this restriction and illustrate the distinction between the nonsingular case and the unrestricted case.

Article information

Source
Rocky Mountain J. Math., Volume 45, Number 5 (2015), 1425-1444.

Dates
First available in Project Euclid: 26 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1453817247

Digital Object Identifier
doi:10.1216/RMJ-2015-45-5-1425

Mathematical Reviews number (MathSciNet)
MR3452221

Zentralblatt MATH identifier
1334.53041

Subjects
Primary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 22E25: Nilpotent and solvable Lie groups

Keywords
Two-step nilpotent Lie algebra and totally geodesic subalgebra

Citation

Decoste, Rachelle C.; Demeyer, Lisa. Totally geodesic subalgebras in 2-step nilpotent Lie algebras. Rocky Mountain J. Math. 45 (2015), no. 5, 1425--1444. doi:10.1216/RMJ-2015-45-5-1425. https://projecteuclid.org/euclid.rmjm/1453817247


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References

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