2-step nilpotent Lie groups of higher rank

The X-ray transform on 2-step nilpotent Lie groups of higher rank

“…Therefore, the methods of [10] do not immediately apply to these spaces. The spaces in [22] differ also significantly from Heisenberg-type groups which do not even infinitesimally have higher rank.…”

“…The main result in this article is about injectivity of the X-ray transform and a support theorem for a certain class of 2-step nilpotent Lie groups with a left invariant metric and higher rank introduced in [22]. By [14] these spaces have conjugate points.…”

“…Let h = R 2q = C q and z = t q−1 ⊂ su(q) ⊂ so(2q) be the Lie algebra of the maximal torus of SU (q) and consider the 2-step nilpotent Lie group N q with Lie algebra n q = z ⊕ h = t q ⊕ R 2q endowed with a left invariant metric. In [22], it was shown that for every q ∈ N, q ≥ 3, the Lie group N q has the property that each geodesic is contained in a totally geodesic immersed 2-dimensional flat submanifold. The reduction principle and Theorem 3.1 yield Theorem 3.2.…”

The Moduli Space of Six-Dimensional Two-Step Nilpotent Lie Algebras

“…The geometry of 2-step nilpotent Lie groups with a left-invariant metric is very rich and has been widely studied since the papers of A. Kaplan [9,10] and P. Eberlein [4] (see, for instance [2,3,5,6,7,12,15] for recent papers on this subject). Important examples of such Lie groups are provided by groups of Heisenberg type [10].…”

“…the minimal nullity of the Jacobi operators. In [15] it was proved that groups of Heisenberg type have infinitesimal rank one. We show that this is also the case for any n (α + ,α − ) ∈ D 6,2 with the exception of n (1,1) ∼ = h 3 ⊕ h 3 and n (1/2,1/2) ∼ = n 5 ⊕ R, both endowed with the product metric, whose rank is two.…”