Geodesic vectors and flat totally geodesic subalgebras of six-dimensional filiform metric Lie algebras

A nilpotent Lie algebra n equipped with an Euclidean inner product is called nilpotent metric Lie algebra. In this paper we describe the sets of the geodesic vectors and the flat totally geodesic subalgebras of the six-dimensional filiform metric Lie algebras. In this class with the exception of the metric Lie algebras corresponding to the standard filiform Lie algebra the flat totally geodesic subalgebras of every metric Lie algebra have dimension at most two.