Abstract
Abstract A metric Lie algebra is a Lie algebra endowed with a Euclidean inner product. A subalgebra is called flat, respectively totally geodesic, if its exponential image in the corresponding Lie group with left invariant Riemannian metric is flat, respectively a totally geodesic submanifold. A non-zero vector is geodesic, if the generated one-dimensional subspace is totally geodesic. We study geodesic vectors and flat totally geodesic subalgebras in two-step nilpotent metric Lie algebras and show that their linear structure is independent of the inner product of the metric Lie algebra. We determine the geodesic vectors and the flat totally geodesic subalgebras in the two-step nilpotent metric Lie algebras of dimension ≤ 6.
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