The geometry of filiform nilpotent Lie groups (Q613533)

The authors study the geometry of a filiform nilpotent Lie group endowed with a left-invariant metric. This is a contribution to the geometry of higher-step nilpotent Lie groups. A filiform is an \(n\)-dimensional nilpotent Lie group for which the lower central series of its corresponding Lie algebra is as long as possible, having \(n-2\) nontrivial subalgebras. That is, an \(n\)-dimensional filiform Lie algebra is \((n-1)\)-step. They describe the connection and curvatures, and they investigate necessary and sufficient conditions for subgroups to be totally geodesic submanifolds. They also classify one-parameter subgroups which are geodesics.