Classification of left-invariant metrics on the Heisenberg group

The index of symmetry of three-dimensional Lie groups with a left-invariant metric

“…It is easy to see that the linear isometry from (h 1 , ·, · ) onto (h 1 , ·, · ′ ) mapping X 1 , X 2 , X 3 into X ′ 1 , X ′ 2 , X 3 respectively, lifts to an isometry from (H 1 , ·, · ) onto (H 1 , ·, · ′ ). Recall that the above argument fails in higher dimensions (see [Vuk15]). In fact, given an inner product ·, · on the (2n + 1)-dimensional Heisenberg Lie algebra h n , we decompose as an orthogonal sum h n = v ⊕ z and for every Z ∈ z we have associated the element j(Z) ∈ so(v) given by j(Z)X, Y = [X, Y ], Z .…”

Geometry of left-invariant Randers metric on the Heisenberg groups

“…In the other hand, every 2-step nilpotent Lie group of odd dimension with a one-dimensional center is locally isomorphic to the Heisenberg group H 2nþ1 . Following [8], any positive definite inner product on H 2nþ1 is given by the following theorem. For σ 1 ≥Á Á Á σ n ≥ 1 and σ 5 (σ 1 , .…”

On the Finsler geometry of the Heisenberg group $H_{2n+1}$ and its extension

“…For example, it is proved that up to homothety, there is a unique left-invariant Riemannian metric on the Heisenberg group H 3 , whereas there are three metrics in the Lorentzian case on these spaces [24]. In [25] this study generalized to Riemannian and Lorentzian metrics on the Heisenberg group H 2n+1 of dimension 2n + 1 and a classification of left-invariant Riemannian and Lorentzian metrics on this group is given.…”

“…Then we apply this result to prove that Heisenberg groups which are equipped with left-invariant Randers metrics of Douglas type can be non-Berwardian generalized Berwald spaces, whereas oscillator groups are generalized Berward spaces which are Berwardian. [25]. By Theorem 3.1 in [25] any left-invariant Riemannian metric on H 2n+1 is given by g σ λ,1 and by [21] with respect to this metric we get the orthonormal basis {x 1 , .…”

“…[25]. By Theorem 3.1 in [25] any left-invariant Riemannian metric on H 2n+1 is given by g σ λ,1 and by [21] with respect to this metric we get the orthonormal basis {x 1 , . .…”