Geometry of left invariant Randers metric on the Heisenberg group

In this paper, we investigate the geometry of left-invariant Randers metrics on the Heisenberg group.


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 [7]). Randers metrics as a special case of Finsler metrics are constructed by Reimannian 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 [8]. Also, curvature properties of Douglas-type Randers metrics on five dimensional two-step homogeneous nilmanifolds can be found in [9].
The Heisenberg groups play a crucial role in theoretical physics, and they are well understood from the viewpoint of sub-Riemannain geometry. These groups arise in the description of onedimensional 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 [3] for a special case of five-dimensional Heisenberg group by investigating the geometry of left-invariant Randers metrics on the Heisenberg group H 2n+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 H 2n+1 have constant negative scalar curvature. Also, we show the Heisenberg group H 2n+1 can not admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, an explicit formula for computing flag curvature of Z-Randers metrics is obtained.

Preliminaries
In this section, we summarize the main concepts and definitions that are needed in this paper.
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 g = T e G. So, the geometry of a left-invariant metric on a Lie group G can be recovered from the geometry of its associated inner product space g = T e G. For instance, let G be a Lie group with a left-invariant metric g then the Koszul formula on U, V, W ∈ g is given by where, ., . denotes the induced inner product on g by g.
The rest of this section is devoted to remind some basic notions on Finsler geometry and in particular to developing the definition 2.1 for Finsler manifolds.
A Finsler metric on a manifold M is a function F : T M → [0, ∞) with the following properties: (i)Regularity: F is smooth on the entire slit tangent bundle T M\{0}.
The Finsler geometry counterpart of the Riemannian sectional curvature is the notion of flag curvature. The flag curvature is defined as follows: is the osculating Riemannain metric, and ∇ is the Chern-Rund connection induced by F on the pull-back bundle π * T M (see [1]).
A special case of Finsler metrics are Randers metrics which are constructed by Riemannaian metrics and vector fields (1-forms). In fact, for a Riemannian metric g and a vector field X on M such that g(X, X) < 1, the Randers metric F , defined by g and X, is a Finsler metric as follows: A Randers metric of the form (2.1) is called 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 the such metrics the Chern connection of Randers metric F coincide with the Levi-civita connection of g. Also, if for a Randers metric of the form (2.1) 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 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 Now, we are prepare to generalize the definition 2.1 for Finsler manifolds.

Definition 2.2 A Finsler metric F on a Lie group G is said to be left-invariant if
where, e is the unit element of G.
Suppose that G is a Lie group, g and U are left-invariant Riemnnian metric and left-invariant vector field on G, respectively, such that g(U, U) < 1. Then the formula (2.1) defines a left-invariant Randers metric on G. In fact, one can easily check that there is a bijective correspondence between left-invariant Randers metrics on a Lie group G with the underlying Reimannian metric g and the left-invariant-vector fields with length < 1. Therefore, the invariant Randers metrics are one-to-one corresponding to the set (see [8], Proposition 3.1) Suppose that G is a Lie group with a left-invariant Randers metric F which is defined by a U ∈ g, then the S-curvature is given by

Geometry of Heisenberg groups
In this section, we investigate the Riemannian geometry of left-invariant metrics on the Heisenberg group H 2n+1 , of dimension 2n + 1.
The Heisenberg group H 2n+1 is defined on the base manifold R 2n × R by multiplication where, ω denotes the standard symplectic form on R 2n . Its associated Lie algebra H 2n+1 is with the following Lie bracket The center of H 2n+1 is one-dimensional, hence the Heisenberg group is 2-step nilpotent. In the other hand, every 2-step nilpotent Lie group of odd dimension with a one-dimensional center is locally isomorphism to the Heisenberg group H 2n+1 .
Following [11], any positive definite inner product on H 2n+1 is given by the following theorem. For σ 1 ≥ · · · σ n ≥ 1 and σ = (σ 1 , · · · , σ n ) denote D n (σ) = diag(σ 1 , σ 1 , · · · , σ n , σ n ) Theorem 3.1 [11] Any positive definite inner product on H 2n+1 , up to the automorphism of Lie algebra (i.e., in some basis of H 2n+1 such that the commutators are given by (3.1 We have already observed above that every positive definite inner product on the Heisenberg algebra H 2n+1 with the commutator specified in (3.1) has a diagonal representation. According to theorem 3.1, we calculate the geometry of a left-invariant metric on the Heisenberg group H 2n+1 .
which is a special form of the Lie bracket given by (3.1). Suppose that denote the corresponding left-invariant vector fields on H 2n+1 . Fix a Riemannain metric g on the Heisenberg group H 2n+1 and denote σ n = 1. Then the Levi-Civita connection of g is given by the following theorem.
Theorem 3.4 The Riemannian curvature tensor of ∇, denoted by R, satisfies the following relations.
where, δ ij is the Kronecker delta.
Proof 3.5 Routine computations show these results. For example we compute the second formula. One can write, Theorem 3.6 The Ricci curvature tensor of ∇, denoted by Ric, satisfies the following relations.

Proof 3.7 It is straightforward to check these results. For instance, we can write:
Proof 3. 9 We have,

Main Results
In this section, our main results will be stated.
then the relations in theorem 3.2 show that a = a i = b i = 0 for 1 ≤ i ≤ n, which is a contradiction.
Note that a Finsler metric is said to be Ricci-quadratic if its Ricci curvature Ric(x, y) is quadratic with respect to y.

is a left-invariant Randers-Douglas metric on H 2n+1 , and F is Ricci-quadratic. Then by ([2], Theorem 7.9) F is Berwald type. But the relations in theorem 3.2 show that Q = 0, which is a contradiction.
We recall that a naturally reductive homogeneous space, is a reductive homogeneous Riemannian Moreover, when H = {e}, then m = g and the above condition can be rewrite as follows Now we are able to prove the following result.
As we observed in theorem 4.1, the Heisenberg group H 2n+1 can not 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 H 2n+1 and the Randers metrics are Z-Randers metrics. We say a left-invariant Randers metric on H 2n+1 is Z-Randers metric if and only if Q ∈ span < Z >. In fact, the condition Q ∈ span < Z > guarantees that the Randers metric is not Berwald.
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 [4].

So, ad(Q) is skew symmetric if and only if
) be a geodesic orbit Finsler space, then a similar argument applies to ad(Q) derives the assertion.
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 φ in the complete group of isometries I(M, F ) such that φ(p) = q. A weakly symmetric Finsler space must be a geodesic orbit Finsler space by Theorem 6.3 in [2]. In the rest of this section, we will restrict our attention to the geometry of Z-Randers metrics on the Heisenberg group H 2n+1 .
Z for a real number 0 < ξ < 1. Using method described in ( [10], Theorem 3.10) and applied in [6] we calculate the osculating metric and the Chern-Rund connection directly. One can easily check the following relations.
The local components of the Chern-Rund connection associated to the osculating metric ., . W with respect to the basis {U i , V i , W } n i=1 which is denoted by ∇ are given by the following formulas.
According to the above relations, for the Riemannian curvature of the Chern-Rund connection ∇ denoted by R, we have Applying the above computations, we find that The above theorem shows that the flag curvature of every Z-Randers metric on the Heisenberg group H 2n+1 with flag pole W = 1 √ λ Z is strictly positive.
Remark 4.14 Considering some special cases of ξ and σ 1 , the above theorem shows that there exist flags of strictly negative and strictly positive curvatures on Heisenberg groups.

Conclusion
In this paper, we investigated the geometry of left-invariant Randers metrics on the Heisenberg group H 2n+1 , of dimension 2n + 1. We determined the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature of a left-invariant metric on H 2n+1 and showed the Heisenberg groups H 2n+1 have constant negative scalar curvature. Other geometric properties of such spaces are investigated. Also, we showed the Heisenberg group H 2n+1 can not 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 H 2n+1 admits a Randers metric of general Dauglus 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 H 2n+1 and giving an explicit formula for computing flag curvature in general case would be a matter of another paper.