Abstract
Purpose
In this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics.
Design/methodology/approach
In the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature.
Findings
In the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.
Originality/value
In this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of left-invarainat Randers metrics will be studied.
Keywords
Citation
Fasihi-Ramandi, G. and Azami, S. (2023), "Geometry of left-invariant Randers metric on the Heisenberg groups", Arab Journal of Mathematical Sciences, Vol. 29 No. 1, pp. 52-62. https://doi.org/10.1108/AJMS-01-2021-0015
Publisher
:Emerald Publishing Limited
Copyright © 2021, Ghodratallah Fasihi-Ramandi and Shahroud Azami
License
Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1. Introduction
Research into left-invariant Riemannian metrics on Lie groups is an active subject of research and this topic is mentioned among many author’s works so far. The curvature properties of such metrics on various kinds of Lie groups are mainly investigated in classical works of Milnor (see [1]).
Randers metrics as a special case of Finsler metrics are constructed by Riemannian metrics and vector fields. Similar to the Riemannian case, the notion of left-invariant Randers metrics on a Lie group G is defined, and the geometry of such spaces is part of many author’s interest topic. A general study of Berwald-type Randers metric on two-step homogeneous nilmanifolds of dimension five is done in [2]. Also, curvature properties of Douglas-type Randers metrics on five dimensional two-step homogeneous nilmanifolds can be found in [3].
The Heisenberg groups play a crucial role in theoretical physics, and they are well understood from the viewpoint of sub-Riemannian geometry. These groups arise in the description of one-dimensional quantum mechanical systems. More generally, one can consider Heisenberg groups associated to n-dimensional systems, and most generally, to any symplectic vector space.
In this study, we develop the results of [4] for a special case of five-dimensional Heisenberg group by investigating the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1 [5]. Considering a left-invariant Randers metric, we give the Levi–Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature. Also, we show the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained.
2. Preliminaries
In this section, we summarize the main concepts and definitions that are needed in this paper.
A (semi-)Riemannian metric g on a Lie group G is said to be left-invariant if
It is well-known that there is a bijective correspondence between left-invariant metrics on a Lie group G, and inner products on its associated Lie algebra
The rest of this section is devoted to remind some basic notions on Finsler geometry and in particular to developing the definition 1 for Finsler manifolds.
A Finsler metric on a manifold M is a function F: TM → [0, ∞) with the following properties:
Regularity: F is smooth on the entire slit tangent bundle TM\{0}.
Positive homogeneity: F(x, λy) = λF(x, y) for all λ > 0.
Strong convexity: The n × n Hessian matrix
The Finsler geometry counterpart of the Riemannian sectional curvature is the notion of flag curvature. The flag curvature is defined as follows:
A special case of Finsler metrics are Randers metrics which are constructed by Riemannian metrics and vector fields (1-forms). In fact, for a Riemannian metric g and a vector field X on M such that
Now, we are prepared to generalize the definition 1 for Finsler manifolds.
A Finsler metric F on a Lie group G is said to be left-invariant if
A Randers metric of the form (1) is called of Berwald type if and only if the vector field X is parallel with respect to Levi–Civita connection of g. It is well-known that in such metrics the Chern-Rund connection of Randers metric F coincides with the Levi–Civita connection of g. Also, if for a Randers metric of the form (1) the 1-form g(X, .) is closed, then the Randers metric is said to be of Douglas type. In the case that M = G is a Lie group, one can easily check that a left-invariant Randers metric on G with underlying left-invariant Riemannian metric is of Douglas type if and only if its underlying vector field Q satisfies the following equation
Suppose that G is a Lie group, g and U are left-invariant Riemannian metric and left-invariant vector field on G, respectively, such that
Suppose that G is a Lie group with a left-invariant Randers metric F which is defined by a
3. Geometry of Heisenberg groups
In this section, we investigate the Riemannian geometry of left-invariant metrics on the Heisenberg group H2n+1, of dimension 2n + 1.
The Heisenberg group H2n+1 is defined on the base manifold
The center of
Following [8], any positive definite inner product on
[8] Any positive definite inner product on
We have already observed above that every positive definite inner product on the Heisenberg algebra
Let β = {u1, v1, …, un, vn, z} is a basis of
The Levi–Civita connection ∇ of the left-invariant metric g, satisfies the following equations.
Straightforward computations using Koszul’s formula, show the above relations, for example we compute the first formula. We have,
Hence,
The Riemannian curvature tensor of ∇, denoted by R, satisfies the following relations.
Routine computations show these results. For example we compute the second formula. One can write,
The Ricci curvature tensor of ∇, denoted by Ric, satisfies the following relations.
It is straightforward to check these results. For instance, we can write:
The above computations indicate that H2n+1 neither cannot be Einstein nor the metric is conformally flat. The next theorem shows that the Heisenberg groups H2n+1 have constant negative scalar curvature.
The scalar curvature of ∇, denoted by R, is given by the following formula.
We have,
4. Main results
In this section, our main results will be stated.
There is not any left-invariant Randers metric of Berwald type on the Heisenberg group
As mentioned before, a left-invariant Randers metric on Lie group G is constructed by a left-invariant vector field with length <1. In the other hand, by definition a Randers metric is Berwald type if and only if the vector field is parallel with respect to the Levi-Civita connection. Suppose that Q is a left-invariant vector field on
Note that a Finsler metric is said to be Ricci-quadratic if its Ricci curvature Ric(x, y) is quadratic with respect to y.
There is not any left-invariant Randers metric of Douglas type on the Heisenberg group H2n+1 which is Ricci-quadratic.
It is worth mentioning that there is a bijective correspondence between left-invariant Randers metric of Douglas type on H2n+1 and the set
We recall that a naturally reductive homogeneous space is a reductive homogeneous Riemannian manifold
Moreover, when H = {e}, then
Now we are able to prove the following result.
Randers Heisenberg group H2n+1 of dimension 2n + 1 is never naturally reductive.
Suppose, conversely, that a Randers metric F on H2n+1 is naturally reductive. Then by theorem 4.1 in [9] the underling Riemannian metric is naturally reductive and we have
If we replace (U, V, W) by (U1, V1, Z), then we have the contradiction λ = 0.
As we observed in theorem 6, the Heisenberg group H2n+1 cannot admit left-invariant Randers metric of Berwald type. A special family of non-Berwald left-invariant Randers metrics which give us a geometric relationship between the Lie algebra
A homogeneous Finsler space (M, F) is said to be a geodesic orbit space if every geodesic in M is an orbit of 1-parameter group of isometries. More details on such spaces can be found in [10].
All Douglas Heisenberg groups H2n+1 are not geodesic orbit spaces. The only Randers metrics on Heisenberg group H2n+1 which could be geodesic orbit are Z-Randers metric.
Suppose that
So, ad(Q) is skew symmetric if and only if ai = bi = 0 for 1 ≤ i ≤ n, which is a contradiction.
Now, let
Recall that a connected Finsler space (M, F) is said to be a weakly symmetric space if for every two points p and q in M there exists an isometry
All Douglas Heisenberg groups H2n+1 are never weakly symmetric spaces.
In the rest of this section, we will restrict our attention to the geometry of Z-Randers metrics on the Heisenberg group H2n+1.
For every Z-Randers metric on the Heisenberg group H2n+1 the flag curvature of the Chern-Rund connection with flag pole
Suppose that
The local components of the Chern-Rund connection associated to the osculating metric 〈.,.〉W with respect to the basis
According to the above relations, for the Riemannian curvature of the Chern-Rund connection
Applying the above computations, we find that
The above theorem shows that the flag curvature of every Z-Randers metric on the Heisenberg group H2n+1 with flag pole
For every Z-Randers metric on the Heisenberg group H2n+1 the flag curvature of the Chern-Rund connection with flag pole W = U1 is given by the following equations.
Suppose that
The local components of the Chern-Rund connection associated to the osculating metric 〈.,.〉W with respect to the basis
According to the above relations, for the Riemannian curvature of the Chern-Rund connection
Applying the above computations, we find that
Considering some special cases of ξ and σ1, the above theorem shows that there exist flags of strictly negative and strictly positive curvatures on Heisenberg groups.
5. Conclusion
In this paper, we investigated the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. We determined the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature of a left-invariant metric on H2n+1 and showed the Heisenberg groups H2n+1 have constant negative scalar curvature. Other geometric properties of such spaces are investigated. Also, we showed the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, by computing flag curvature it was shown that there exist flags of strictly negative and strictly positive curvatures.
The question of whether the Heisenberg group H2n+1 admits a Randers metric of general Douglas type is not considered in this paper. Also, we have computed the flag curvature of a special kind of Randers metric (namely Z-Randers metric) on H2n+1 and giving an explicit formula for computing flag curvature in the general case would be matter of another paper.
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