Geometry of left-invariant Randers metric on the Heisenberg groups

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 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.

Findings

In the second part, we present our main results. We show that the Heisenberg group H_2n+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.

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

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 H_2n+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 H_2n+1 have constant negative scalar curvature. Also, we show the Heisenberg group H_2n+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.

Definition 1.

A (semi-)Riemannian metric g on a Lie group G is said to be left-invariant if La*(g)=g, for all a ∈ G.

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=TeG. 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=TeG. 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

2〈∇UV,W〉=〈[U,V],W〉−〈[V,W],U〉+〈[W,U],V〉,

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 1 for Finsler manifolds.

A Finsler metric on a manifold M is a function F: TM → [0, ∞) with the following properties:

1. Regularity: F is smooth on the entire slit tangent bundle TM\{0}.

2. Positive homogeneity: F(x, λy) = λF(x, y) for all λ > 0.

3. Strong convexity: The n × n Hessian matrix

[gij]=[12∂2F2∂yi∂yj]

is positive definite at every point (x, y) ∈ TM\{0}.

The Finsler geometry counterpart of the Riemannian sectional curvature is the notion of flag curvature. The flag curvature is defined as follows:

K(P,Y)=gY(R(X,Y)Y,X)gY(Y,Y)gY(X,X)−gY2(Y,X)

where,

gY(X,Z)=12∂2∂s∂t|s=t=0F2(Y+sX+tZ),

is the osculating Riemannian metric, P = span{X, Y}, R(X, Y)(Z) = ∇_X∇_YZ −∇_Y∇_XZ −∇_[X,Y] Z and ∇ is the Chern-Rund connection induced by F on the pull-back bundle π*TM (see [6]).

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 g(X,X)<1, the Randers metric F, defined by g and X, is a Finsler metric as follows:

(1)F(x,y)=g(y,y)+g(X(x),y)∀x∈M,y∈TxM.

Now, we are prepared to generalize the definition 1 for Finsler manifolds.

Definition 2.

A Finsler metric F on a Lie group G is said to be left-invariant if

F(a,v)=F(e,La−1*(v))∀a∈G,v∈TaG,

where e is the unit element of G.

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

<[U,V],Q>=0,∀U,V∈g=TeG.

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 g(U,U)<1. Then Formula (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 Riemannian 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 [2], Proposition 3.1)

{U∈g|<U,U><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

S(e,v)=n+12〈[U,v],〈v,U〉U−v〉F(e,v)−〈[U,v],U〉,

and the Randers metric has vanishing S-curvature if and only if the linear endomorphism ad(U) of g is skew symmetric with respect to the inner product 〈., .〉 on the Lie algebra g. (See [7] Proposition 7.5).

3. 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 R2n×R by multiplication

(x,λ).(y,μ)=(x+y,λ+μ+ω(x,y)),

where ω denotes the standard symplectic form on R2n. Its associated Lie algebra H2n+1 is

H2n+1=R2n⊕R={(x,λ)|x∈R2n,λ∈R},

with the following Lie bracket

(2)[(x,λ),(y,μ)]=(0,ω(x,y)).

The center of H2n+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 isomorphic to the Heisenberg group H_2n+1.

Following [8], any positive definite inner product on H2n+1 is given by the following theorem. For σ₁ ≥⋯ σ_n ≥ 1 and σ = (σ₁, …, σ_n) denote

Dn(σ)=diag(σ1,σ1,…,σn,σn).

Theorem 1.

[8] Any positive definite inner product on H2n+1, up to the automorphism of Lie algebra (i.e. in some basis of H2n+1 such that the commutators are given by (2)) is represented by the matrix

Dn−1(σ)00S

where S = diag(1, 1, λ), λ > 0.

We have already observed above that every positive definite inner product on the Heisenberg algebra H2n+1 with the commutator specified in (2) has a diagonal representation. According to theorem 1, we calculate the geometry of a left-invariant metric on the Heisenberg group H_2n+1.

Let β = {u₁, v₁, …, u_n, v_n, z} is a basis of H2n+1 with non-zero commutators

[ui,vi]=z,i=1,…,n,

which is a special form of the Lie bracket given by (2). Suppose that

U1,V1,…,Un,Vn,Z

denote the corresponding left-invariant vector fields on H_2n+1. Fix a Riemannian 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 2.

The Levi–Civita connection ∇ of the left-invariant metric g, satisfies the following equations.

∇UiVi=12  Z,∇ViUi=−12  Z,∇UiZ=−λ2σi Vi,∇ZUi=−λ2σi Vi,∇ViZ=λ2σi Ui,∇ZVi=λ2σi Ui.

Proof.

Straightforward computations using Koszul’s formula, show the above relations, for example we compute the first formula. We have,

2〈∇UiVi,Uj〉=〈[Ui,Vi],Uj〉−〈[Vi,Uj],Ui〉+〈[Uj,Ui],Vi〉=0,2〈∇UiVi,Vj〉=〈[Ui,Vi],Vj〉−〈[Vi,Vj],Ui〉+〈[Vj,Ui],Vi〉=0,2〈∇UiVi,Z〉=〈[Ui,Vi],Z〉−〈[Vi,Z],Ui〉+〈[Z,Ui],Vi〉=〈Z,Z〉=λ.

Hence, ∇UiVi=12Z.

Theorem 3.

The Riemannian curvature tensor of ∇, denoted by R, satisfies the following relations.

R(Ui,Uk)Uk=0,R(Ui,Vk)Vk=−3λ4σk δikUk,R(Ui,Z)Z=λ24σi2 Ui,R(Vi,Uk)Uk=−3λ4σk δikVk,R(Vi,Vk)Vk=0,R(Vi,Z)Z=λ24σi2 Vi,R(Z,Uk)Uk=λ4σk Z,R(Z,Vk)Vk=λ4σk  Z,

where δ_ij is the Kronecker delta.

Proof.

Routine computations show these results. For example we compute the second formula. One can write,

R(Ui,Vk)Vk=∇Ui∇VkVk−∇Vk∇UiVk−∇[Ui,Vk]Vk=−δik(12∇VkZ+∇ZVk)=−3λ4σkδikUk.

Theorem 4.

The Ricci curvature tensor of ∇, denoted by Ric, satisfies the following relations.

Ric(Ui,Uj)=−λδij2σi,Ric(Vi,Vj)=−λδij2σi,Ric(Z,Z)=λ22∑k=1n1σk2.

Proof.

It is straightforward to check these results. For instance, we can write:

Ric(Vi,Vj)=1λ〈R(Vi,Z)Z,Vj〉+∑k=1n1σk〈R(Vi,Uk)Uk,Vj〉=1λ〈λ24σi2 Vi,Vj〉+∑k=1n1σk〈−3λ4σk δikVk,Vj〉=λ4σi2〈Vi,Vj〉−3λ4σi2〈Vi,Vj〉=−λ2σi2〈Vi,Vj〉=−λ2σi2 δijσi=−λδij2σi.

Remark.

The above computations indicate that H_2n+1 neither cannot be Einstein nor the metric is conformally flat. The next theorem shows that the Heisenberg groups H_2n+1 have constant negative scalar curvature.

Theorem 5.

The scalar curvature of ∇, denoted by R, is given by the following formula.

R=−λ2∑i=1n1σi2.

Proof.

We have,

R=1λRic(Z,Z)+∑i=1n1σiRic(Ui,Ui)+∑i=1n1σiRic(Vi,Vi)=∑i=1nλ21σi2−λ21σi2−λ21σi2=−λ2∑i=1n1σ2.

4. Main results

In this section, our main results will be stated.

Theorem 6.

There is not any left-invariant Randers metric of Berwald type on the Heisenberg group H2n+1.

Proof.

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 H2n+1 which is parallel with respect to the Levi-Civita connection. This implies that for all U∈H2n+1 we have ∇_UQ = 0. Let

Q=aZ+∑i=1naiUi+biVi,

then the relations in theorem 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.

Corollary 1.

There is not any left-invariant Randers metric of Douglas type on the Heisenberg group H_2n+1 which is Ricci-quadratic.

Proof.

It is worth mentioning that there is a bijective correspondence between left-invariant Randers metric of Douglas type on H_2n+1 and the set {U∈H2n+1|<U,U><1,<U,Z>=0}. Suppose that Q=∑i=1naiUi+biVi is a left-invariant vector field such that

F(a,v)=g(v,v)+g(Q(a),v),

is a left-invariant Randers-Douglas metric on H_2n+1, and F is Ricci-quadratic. Then by ([7], Theorem 7.9) F is Berwald type. But the relations in theorem 2 show that Q = 0, which is a contradiction.

We recall that a naturally reductive homogeneous space is a reductive homogeneous Riemannian manifold M=GH with a decomposition g=h+m, that satisfies

〈[X,Y]m,Z〉=−〈Y,[X,Z]m〉,X,Y,Z∈m.

Moreover, when H = {e}, then m=g and the above condition can be rewrite as follows:

〈[X,Y],Z〉+〈Y,[X,Z]〉=0,X,Y,Z∈g.

Now we are able to prove the following result.

Corollary 2.

Randers Heisenberg group H_2n+1 of dimension 2n + 1 is never naturally reductive.

Proof.

Suppose, conversely, that a Randers metric F on H_2n+1 is naturally reductive. Then by theorem 4.1 in [9] the underling Riemannian metric is naturally reductive and we have

〈[U,V],W〉+〈V,[U,W]〉=0,U,V,W∈H2n+1.

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 H_2n+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 H2n+1 and the Randers metrics are Z-Randers metrics. We say a left-invariant Randers metric

(3)F(a,v)=g(v,v)+g(Q(a),v),

on H_2n+1 is a 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 [10].

Theorem 7.

All Douglas Heisenberg groups H_2n+1 are not geodesic orbit spaces. The only Randers metrics on Heisenberg group H_2n+1 which could be geodesic orbit are Z-Randers metric.

Proof.

Suppose that Q=∑i=1naiUi+biVi is a left-invariant vector field such that

F(a,v)=g(v,v)+g(Q(a),v),

is a left-invariant Randers-Douglas metric on H_2n+1. If (H_2n+1, F) is a geodesic orbit Finsler space, then by Corollary 5.3 of [10], the S-curvature vanishes which means ad(Q) is a skew symmetric on the H2n+1. But we have

[Q,Ui]=−biZ,[Q,Vi]=aiZ,[Q,Z]=0.

So, ad(Q) is skew symmetric if and only if a_i = b_i = 0 for 1 ≤ i ≤ n, which is a contradiction.

Now, let F(a,v)=g(v,v)+g(Q(a),v) is a left-invariant Randers metric on H_2n+1 with Q=aZ+∑i=1naiUi+biVi. If (H_2n+1, F) is a geodesic orbit Finsler space, then a similar argument applied 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 [7].

Corollary 3.

All Douglas Heisenberg groups H_2n+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 H_2n+1.

Theorem 8.

For every Z-Randers metric on the Heisenberg group H_2n+1 the flag curvature of the Chern-Rund connection with flag pole W=1λZ is given by the following equation.

K(Π,W)=K(Π)=λ4∑i=1n(ai2+bi2)σi∑i=1n(ai2+bi2)σi,

where, Π = span{U, W} and U=∑i=1naiUi+biVi.

Proof.

Suppose that F(a,v)=g(v,v)+g(Q(a),v) is a Z-Randers metric on H_2n+1 with Q=ξW=ξλZ for a real number 0 < ξ < 1. Using the method described in ([11], Theorem 3.10) which is applied in [12] (for a special frame, namely, Berwald-Moor frame), we calculate the osculating metric and the Chern-Rund connection directly. One can easily check the following relations.

〈Ui,Uj〉W=(1+ξ)δijσi,〈Vi,Vj〉W=(1+ξ)δijσi,〈Ui,Vj〉W=0,〈Ui,W〉W=0,〈Vi,W〉W=0,〈W,W〉W=(1+ξ)2.

The local components of the Chern-Rund connection associated to the osculating metric 〈.,.〉_W with respect to the basis {Ui,Vi,W}i=1n which is denoted by ∇¯ are given by the following formulas.

∇¯UiVi=λ2W,∇¯ViUi=−λ2W,∇¯UiW=−λ2σi(1+ξ)Vi,∇¯WUi=−λ2σi(1+ξ)Vi,∇¯ViW=λ2σi(1+ξ)Ui,∇¯WVi=λ2σi(1+ξ)Ui.

According to the above relations, for the Riemannian curvature of the Chern-Rund connection ∇¯ denoted by R¯, we have

R¯(Ui,W)(W)=λ4σi2 (1+ξ)2Ui,R¯(Vi,W)(W)=λ4σi2 (1+ξ)2Vi.

Applying the above computations, we find that

K(Π,W)=〈R¯(Ui,W)W,Ui〉W〈W,W〉W〈Ui,Ui〉W−〈W,Ui〉W2=λ4σi2,K(P,W)=〈R¯(Vi,W)W,Vi〉W〈W,W〉W〈Vi,Vi〉W−〈W,Vi〉W2=λ4σi2,

where, Π = span{U_i, W} and P = span{V_i, W}. Now, let U=∑i=1naiUi+biVi and Π = span{U, W}, then easy calculations show

K(Π,W)=K(Π)=λ4∑i=1n(ai2+bi2)σi∑i=1n(ai2+bi2)σi.

The above theorem shows that the flag curvature of every Z-Randers metric on the Heisenberg group H_2n+1 with flag pole W=1λZ is strictly positive.

Theorem 9.

For every Z-Randers metric on the Heisenberg group H_2n+1 the flag curvature of the Chern-Rund connection with flag pole W = U₁ is given by the following equations.

K(Π,W)=ξ2λ4σ2,K(P,W)=λ2(1+ξ2)24σ12λ(1+ξ2(1−σ12)),

where, Π = span{U₂, W} and P = span{Z, W}.

Proof.

Suppose that F(a,v)=g(v,v)+g(Q(a),v) is a Z-Randers metric on H_2n+1 with Q=ξλZ for a real number 0 < ξ < 1. If W = U₁ then, one can easily check the following relations.

〈Ui,Uj〉W=δijσi,〈Vi,Vj〉W=δijσi,〈Ui,Vj〉W=0,〈Ui,Z〉W=ξλδ1iσi,〈Vi,Z〉W=0,〈Z,Z〉W=λ(1+ξ2).

The local components of the Chern-Rund connection associated to the osculating metric 〈.,.〉_W with respect to the basis {Ui,Vi,Z}i=1n which is denoted by ∇¯ are given by the following formulas.

∇¯UiVj=12δijZ+ξλ2∑k=1n(δijδ1k+δ1iδjkσiσk)Uk=δijZ+∇¯VjUi,∇¯UiUj=−ξλ2∑k=1n(δ1iδjkσiσk+δ1jδikσjσk)Vk=∇¯UjUi,∇¯UiZ=−λ2σi(1+ξ2)Vi=∇¯ZUi,∇¯ViZ=λ2σi(1+ξ2)Ui=∇¯ZVi,

According to the above relations, for the Riemannian curvature of the Chern-Rund connection ∇¯ denoted by R¯, we have

R¯(U2,W)(W)=ξ2λ4σ1σ2U2,R¯(Z,W)(W)=λ4σ1(1+ξ2)Z.

Applying the above computations, we find that

K(Π,W)=〈R¯(U2,W)W,U2〉W〈W,W〉W〈U2,U2〉W−〈W,U2〉W2=ξ2λ4σ2,K(P,W)=〈R¯(Z,W)W,Z〉W〈W,W〉W〈Z,Z〉W−〈W,Z〉W2=λ2(1+ξ2)24σ12λ(1+ξ2(1−σ12)),

where, Π = span{U₂, W} and P = span{Z, W}.

Remark 1.

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 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 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 H_2n+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 H_2n+1 and giving an explicit formula for computing flag curvature in the general case would be matter of another paper.