DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 603–610

POISSON MANIFOLDS OF GEODESIC ARCS

LUBOMÍR KLAPKA

Abstract. In the present paper a notion of a Poisson manifold of geodesic arcs is introduced and studied. Local coordinate expressions are found. A relation between Poisson manifolds of geodesic arcs and the classical Lagrangian mechanics is clarified.

Definitions of fundamental notions

Let us consider a smooth and finite-dimensional manifold X. A Poisson algebra over X is a Lie algebra structure { , } on C^{∞}(X) that satisfies the Leibniz condition {F G, H} = F {G, H} + G{F, H}. A Poisson bracket { , } is the Lie bracket of the corresponding Poisson algebra. A Casimir function in a Poisson algebra over X is a function F ∈ C^{∞}(X) such that {F, G} = 0 for all functions G ∈ C^{∞}(X). A Poisson manifold is a smooth manifold X with a Poisson algebra over X (Lichnerowicz [1], Weinstein [2], and others). A Poisson mapping is a homomorphism of Poisson manifolds. A Poisson submanifold is a submanifold Y in a Poisson manifold X with a Poisson algebra for which the inclusion is a Poisson mapping. Note that any open submanifold in a Poisson manifold is the Poisson submanifold. A Poisson manifold is Abelian if and only if the corresponding Poisson algebra is Abelian. A Poisson manifold is a symplectic manifold if and only if each of its Casimir functions is a constant function.

Let us consider a symmetric linear connection Γ on X and a closed interval ⟨0, 1⟩ ⊂ ℝ. A geodesic arc γ of the connection Γ is a smooth mapping γ : ⟨0, 1⟩ → X that is a geodesic of connection Γ. Let W (X) be a set of all geodesic arcs of the connection Γ. It is known (see, e.g., Kobayashi and Nomizu [3]) that there exists a bijective mapping

(1)  β : W (X) ∋ γ → γ̇(0) ∈ V,

1991 Mathematics Subject Classification. 17B65, 53B05, 53B20, 53C05, 53C22, 58F05, 58F99, 70D10, 70G99, 70H35.

Key words and phrases. Poisson manifolds, symmetric linear connections, geodesic arcs, Euler-Lagrange equations.

This paper is final form and no version of it will be submitted for publication elsewhere.

Research supported by grant No. 201/93/2245 from the Czech Grant Agency.

where γ̇ : ⟨0, 1⟩ → T X is the prolongation of geodesic arc γ on the tangent bundle T X and V is an open subset of T X. The set W (X) equipped with a structure of smooth manifold such that β is a diffeomorphism is called a manifold of geodesic arcs.

Let us consider two manifolds of geodesic arcs W (X) and W (Y). Any smooth mapping f : X → Y such that γ ∈ W (X) implies f ∘γ ∈ W (Y) is called a geodesic mapping. The geodesic mapping f defines a mapping L_{f} : W (X) ∋ γ → f ∘ γ ∈ W (Y) of manifolds of geodesic arcs. Evidently, if X is a fibred manifold over the basis Y with the projection f then W (X) is a fibred manifold over the basis W (Y) with projection L_{f}. In such a case W (X) is called a fibred manifold of geodesic arcs L_{f} : W (X) → W (Y). Similarly, if Y is a submanifold of the manifold X with the inclusion f then W (Y) is a submanifold of W (X) with the inclusion L_{f}. In such a case Y is called a geodesic submanifold of the manifold X.

Let us consider a monoid M of all polynomial mappings ⟨0, 1⟩ → ⟨0, 1⟩ whose degree is less than two with the multiplication M × M ∋ (μ, ν) → μ ∘ ν ∈ M. The global chart M ∋ μ → (μ(0), μ(1)) ∈ ⟨0, 1⟩ × ⟨0, 1⟩ defines a structure of a smooth manifold with corners on M in such a way that the multiplication is smooth. The monoid M with the above defined structures will be called the Lie monoid M. Let W (X) is a manifold of geodesic arcs. It is known (see, e.g., [3], [5]) that γ ∈ W (X), μ ∈ M implies γ ∘ μ ∈ W (X) and so the Lie monoid M acts on W (X) from right. Since a mapping V × ⟨0, 1⟩ ∋ (a, b) → β^{−1}(a)(b) ∈ X is smooth, we see that the action R : W (X) × M ∋ (γ, μ) → γ ∘ μ ∈ W (X) is smooth too. We shall use a partial mapping

(2)  R_{μ} : W (X) ∋ γ → γ ∘ μ ∈ W (X).

The action R of the Lie monoid M is of fundamental importance in the theory of geodesic arcs. It can be shown (Whitehead [4], Klapka [5]) that the notion of a symmetric linear connection may be defined with the help of (2).

Definition. Let W (X) be a manifold of geodesic arcs. A Poisson right M-algebra over W (X) is a Poisson algebra over W (X) such that for each μ ∈ M a mapping C^{∞}(W (X)) ∋ \sqrt{F} ∘ R_{μ} ∈ C^{∞}(W (X)) is an endomorphism. A Poisson manifold of geodesic arcs is a manifold of geodesic arcs equipped with a Poisson right Malgebra.

Local expressions

Local coordinate expressions of Poisson manifolds of geodesic arcs can be found in the proof of Theorem 1. Throughout all the following text the standard summation convention with the summation from 1 to dim X is used.

Theorem 1. Let W (X) be a manifold of geodesic arcs of symmetric linear connection Γ on X. Then there exists a bijective correspondence between the set of all Poisson right M-algebras over W (X) and the set of all ordered pairs (g, h) of tensor fields on X satisfying the following three conditions:

a) g is a tensor field of the type (2, 1),

b) h is a tensor field of the type (2, 0),

POISSON MANIFOLDS OF GEODESIC ARCS 605

c) in each chart x ∈ At(X) the components g_{k}^{ij}, h^{ij} satisfy the relations:

h^{ij} + h^{ji} = 0, (3) h^{il}h_{l}^{jk} + h^{jl}h_{l}^{ki} + h^{kl}h_{l}^{ij} = 0, (4) g_{m}^{il}g_{l}^{jk} − g_{m}^{jl}g_{l}^{ik} + g_{ml}^{ik}h^{lj} − g_{ml}^{jk}h^{li} = g_{m}^{lk}h_{l}^{ij} + R_{mlr}^{k}h^{li}h^{rj}, (5) g_{k}^{ij} − g_{k}^{ji} = h_{k}^{ij}, (6) g_{kl}^{ij} + g_{lk}^{ij} + g_{kl}^{ji} + g_{lk}^{ji} = (R_{klm}^{i} + R_{lkm}^{i})h^{mj} + (R_{klm}^{j} + R_{lkm}^{j})h^{mi}, (7)

(8)  \begin{matrix}\frac{1}{2}(h_{lm}^{ij} + h_{ml}^{ij} + R_{lmk}^{i}h^{kj} + R_{mlk}^{i}h^{kj} − R_{lmk}^{j}h^{ki} − R_{mlk}^{j}h^{ki})_{r} \\ +\frac{1}{2}(h_{mr}^{ij} + h_{rm}^{ij} + R_{mrk}^{i}h^{kj} + R_{rmk}^{i}h^{kj} − R_{mrk}^{j}h^{ki} − R_{rmk}^{j}h^{ki})_{l} \\ +\frac{1}{2}(h_{rl}^{ij} + h_{lr}^{ij} + R_{rlk}^{i}h^{kj} + R_{lrk}^{i}h^{kj} − R_{rlk}^{j}h^{ki} − R_{lrk}^{j}h^{ki})_{m} \\ = (R_{mrk}^{i} + R_{rmk}^{i})g_{l}^{kj} + (R_{rlk}^{i} + R_{lrk}^{i})g_{m}^{kj} + (R_{lmk}^{i} + R_{mlk}^{i})g_{r}^{kj} \\ −(R_{mrk}^{j} + R_{rmk}^{j})g_{l}^{ki} − (R_{rlk}^{j} + R_{lrk}^{j})g_{m}^{ki} − (R_{lmk}^{j} + R_{mlk}^{j})g_{r}^{ki}, \end{matrix}

where R_{jkl}^{i} are components of the Riemannian tensor field of the connection Γ and the lower indices which does not belong to the indexation of tensors g, h denote the corresponding covariant derivatives. Proof. For each x we consider a mapping

(9)  ψ : dom ψ ∋ γ → (x ∘ γ(t), \frac{d}{dt}(x ∘ γ(t)))|_{t=0}∈ rng ψ ⊂ ℝ^{2 dim X},

where dom ψ = {γ ∈ W (X) | γ(0) ∈ dom x}. Since ψ ∘ β^{−1}|_{β(dom ψ)}∈ At(V),

the set of all charts of type (9) makes an atlas At(W (X)). The coordinates of the chart ψ ∈ At(W (X)) will be denoted by q^{i}, v^{i}.

Let us consider two tensor fields g and h on X satisfying the conditions a)–c). Let ϑ be the projection dom ψ ∋ γ → γ(0) ∈ dom x and let Γ_{jk}^{i} be the components of the connection Γ. Expressions {q^{i}, q^{j}} = h^{ij} ∘ ϑ, (10) {q^{i}, v^{j}} = ((g_{k}^{ij} − Γ_{kl}^{j}h^{il}) ∘ ϑ)v^{k}, (11)

(12)  \begin{matrix}\{v^{i}, v^{j}\} = ((−\frac{1}{2}h_{lm}^{ij} − \frac{1}{2}R_{lmk}^{i}h^{kj} + \frac{1}{2}R_{lmk}^{j}h^{ki} \\ − Γ_{lk}^{i}g_{m}^{kj} + Γ_{lk}^{j}g_{m}^{ki} + Γ_{kl}^{i}Γ_{rm}^{j}h^{kr}) ∘ ϑ)v^{l}v^{m} \end{matrix}

are components of a skew-symmetric tensor field on W (X). This fact makes one possible to define a skew-symmetric algebra structure C^{∞}(W (X))×C^{∞}(W (X)) ∋ (F, G) → {F, G} ∈ C^{∞}(W (X)) by local coordinate expressions

(13)  \begin{matrix}\{F, G\}|_{dom ψ}= \{q^{i}, q^{j}\}\frac{∂F}{∂q^{i}} \frac{∂G}{∂q^{j}} \\ + \{q^{i}, v^{j}\}(\frac{∂F}{∂q^{i}} \frac{∂G}{∂v^{j}} − \frac{∂F}{∂v^{j}} \frac{∂G}{∂q^{i}} ) + \{v^{i}, v^{j}\}\frac{∂F}{∂v^{i}} \frac{∂G}{∂v^{j}} , \end{matrix}

where ψ ∈ At(W (X)).

Using (6)–(8) we get

(14)  ∂_{ξ}{F, G} = {∂_{ξ}F, G} + {F, ∂_{ξ}G},

where a vector field ξ is defined on W (X) by expressions

(15)  ξ|_{dom ψ}= v^{i} \frac{∂}{∂q^{i}} − (Γ_{jk}^{i} ∘ ϑ)v^{j}v^{k} \frac{∂}{∂v^{i}}

and ∂_{ξ} is the Lie derivative with respect to ξ.

Let us denote 𝔍(F, G, H) = {F, {G, H}} + {G, {H, F }} + {H, {F, G}}. From (3)–(5) we get 𝔍(q^{i}, q^{j}, q^{k}) = 𝔍(q^{i}, q^{j}, v^{k}) = 0 on dom ψ. Consequently for all H ∈ C^{∞}(dom ψ) it holds 𝔍(q^{i}, q^{j}, H) = 0. By the Lie differentiation with respect to ξ we obtain 𝔍(v^{i}, q^{j}, H) + 𝔍(q^{i}, v^{j}, H) = 0. Putting H = v^{k} we can compute 𝔍(q^{i}, v^{j}, v^{k}) = 0. From the above relations for all G, H ∈ C^{∞}(dom ψ) it holds 𝔍(q^{i}, G, H) = 0. By the next Lie derivative with respect to ξ we have 𝔍(v^{i}, G, H) = 0. For all F, G, H ∈ C^{∞}(dom ψ) this implies 𝔍(F, G, H) = 0. Hence there exists a chart ψ ∈ At(W (X)) at each arc of W (X), the Jacobi identity is fulfilled on all C^{∞}(W (X)) and so C^{∞}(W (X)) is a Lie algebra. Besides, the Leibniz condition is fulfilled too and so C^{∞}(W (X)) is a Poisson algebra over W (X).

Let us consider a Poisson algebra over V such that β is a Poisson mapping. Then there exists a Poisson structure on T X such that the inclusion V ⊂ T X and all the mappings λ_{p} : T X ∋ a → pa ∈ T X, where p ∈ ℝ, are Poisson mappings. Furthermore there exists a vector field η on T X such that F ∈ C^{∞}(T X), p ∈ ℝ implies ∂_{ξ}(F ∘ β) = (∂_{η}F) ∘ β, ∂_{η}(F ∘ λ_{p}) = (∂_{η}F) ∘ λ_{p}. According to (14), for each p ∈ ℝ the flow exp(pη) is a Poisson mapping. Thence for each μ ∈ M the mapping R_{μ} = β^{−1} ∘ λ_{μ(1)−μ(0)} ∘ exp(μ(0)η) ∘ β is a Poisson mapping and C^{∞}(W (X)) is a Poisson right M-algebra over W (X).

Conversely, let C^{∞}(W (X)) be a Poisson right M-algebra over W (X). Then (13) holds (see, e.g., [1], [2]). If we sum up the derivatives of the relation {F ∘ R_{μ}, G ∘ R_{μ}} = {F, G} ∘ R_{μ} with respect to μ(0) and with respect to μ(1) then putting μ = id_{⟨0,1⟩} we see that (14) holds too.

Let k ∈ ⟨0, 1⟩, κ : ⟨0, 1⟩ ∋ τ → kτ ∈ ⟨0, 1⟩. For Poisson mapping R_{κ} relations q^{i} ∘ R_{κ} = q^{i}, v^{i} ∘ R_{κ} = kv^{i} hold. Thence

\frac{∂^{2}\{q^{i}, v^{j}\}}{∂v^{r}∂v^{s}} = lim_{k→0} \frac{∂^{2}\{q^{i}, v^{j}\}}{∂v^{r}∂v^{s}} = lim_{k→0} \frac{1}{k} \frac{∂^{2}\{q^{i}, kv^{j}\}}{∂v^{r}∂v^{s}}

= lim_{k→0} \frac{1}{k} \frac{∂^{2}\{q^{i} ∘ R_{κ}, v^{j} ∘ R_{κ}\}}{∂v^{r}∂v^{s}} = lim_{k→0} \frac{1}{k} \frac{∂^{2}\{q^{i}, v^{j}\} ∘ R_{κ}}{∂v^{r}∂v^{s}}

= lim_{k→0} k \frac{∂^{2}\{q^{i}, v^{j}\}}{∂v^{r}∂v^{s}} ∘ R_{κ} = 0.

Similarly

\frac{∂\{q^{i}, q^{j}\}}{∂v^{r}} = lim_{k→0} \frac{∂\{q^{i} ∘ R_{κ}, q^{j} ∘ R_{κ}\}}{∂v^{r}} = lim_{k→0} \frac{∂\{q^{i}, q^{j}\} ∘ R_{κ}}{∂v^{r}}

= lim_{k→0} k \frac{∂\{q^{i}, q^{j}\}}{∂v^{r}} ∘ R_{κ} = 0.

Hence further

lim_{k→0}{q^{i}, v^{j}} ∘ R_{κ} = lim_{k→0}{q^{i}, kv^{j}} = lim_{k→0} k{q^{i}, v^{j}} = 0,

we obtain (10) and \{q^{i}, v^{j}\} = (\overline{g}_{k}^{ij} ∘ ϑ)v^{k} where \overline{g}_{k}^{ij} are functions on dom x. According to (13), the functions h^{ij} and g_{k}^{ij} = \overline{g}_{k}^{ij} + Γ_{kl}^{j}h^{il} are components of tensor fields g and h on X such that (11) holds. The fields g and h satisfy conditions a) and b).

It remains to show that c) holds. From the first Lie derivative of (11) and the lowest three Lie derivatives of (10) with respect to ξ we obtain (7), (6), (12) and (8). Relation (3) is a consequence of (10). Relations (4) and (5) follow from Jacobi identities 𝔍(q^{i}, q^{j}, q^{k}) = 𝔍(q^{i}, q^{j}, v^{k}) = 0. This completes the proof.

Example. In-out manifolds defined in work (Klapka [6]) are isomorphic to Poisson manifolds of geodesic arcs whose Poisson submanifold of constant arcs is Abelian. In such a case relation (10) implies h = 0. Then by (5) and (6) the tensor field g is the field of structure tensors of fibration of cotangent commutative associative algebras over basis X.

The relation to classical Lagrangian mechanics

For a geometric formulation of the first order Lagrangian theory the formalism of fibred manifolds and theirs lower two prolongations is used (see, e.g., Krupka [7]). In case of mechanics we consider a fibred manifold π : ℝ × X → ℝ, where ℝ is the manifold of real numbers and X is a smooth finite-dimensional manifold, with its prolongations J^{1}(ℝ×X), J^{2}(ℝ×X) and projections π_{1} : J^{1}(ℝ×X) → ℝ, π_{21} : J^{2}(ℝ × X) → J^{1}(ℝ × X), π_{20} : J^{2}(ℝ × X) → ℝ × X. We shall use the fiber charts with coordinates t, x^{i}, ẋ^{i}, ẍ^{i}. A time evolution of mechanical system is described by local sections of the fibred manifold π : ℝ × X → ℝ which are solutions of Euler–Lagrange equations

(16)  ϵ_{i} ∘ j^{2}σ = 0.

Here Euler–Lagrange expressions ϵ_{i} are components of π_{20}-horizontal 1-contact differential 2-form on J^{2}(ℝ × X) such that a Lagrange function L satisfying

(17)  ϵ_{i} = \frac{d}{dt}(\frac{∂L}{∂ẋ^{i}} ) − \frac{∂L}{∂x^{i}} ,

where d/dt is the total derivative, locally exists. The Lagrange function is the component of the π_{1}-horizontal differential 1-form on J^{1}(ℝ × X). Note that by the usual notation practice in the classical mechanics all necessary projections are omitted (for instance, the restriction of projection π_{21} absents in the second term of the right-hand side of (17)). We suppose that the Euler–Lagrange expressions are regular, that is, for each point of J^{2}(ℝ × X)

(18)  det(\frac{∂ϵ_{k}}{∂ẍ^{l}} ) ≠ 0.

Let us consider a set E(ℝ × X) of all solutions σ_{ab} : ⟨a, a + b⟩ → ℝ × X of equations (16), where a, b ∈ ℝ, b > 0. Then there exists a bijection

(19)  χ : E(ℝ × X) ∋ σ_{ab} → (b, j^{1}σ_{ab}(a)) ∈ U,

where U is an open subset of the manifold (0, ∞) × J^{1}(ℝ × X). The Poisson structure of this manifold can be found by the methods of classical mechanics. From classical canonical relations

\{x^{i}, x^{j}\} = 0, \{x^{i}, \frac{∂L}{∂ẋ^{j}} \} = δ_{j}^{i}, \{\frac{∂L}{∂ẋ^{i}} , \frac{∂L}{∂ẋ^{j}} \} = 0

we obtain

(20)  \begin{matrix}\{x^{k}, x^{l}\} = 0, \\ \{x^{k}, ẋ^{m}\}\frac{∂ϵ_{l}}{∂ẍ^{m}} = δ_{l}^{k}, \\ \{ẋ^{k}, ẋ^{l}\}\frac{∂ϵ_{r}}{∂ẍ^{k}} \frac{∂ϵ_{s}}{∂ẍ^{l}} = \frac{1}{2}(\frac{∂ϵ_{s}}{∂ẋ^{r}} − \frac{∂ϵ_{r}}{∂ẋ^{s}} ). \end{matrix}

Note that in the classical mechanics the coordinates on ℝ and (0, ∞) are considered as Casimir functions. The set E(ℝ×X) equipped with a Poisson algebra such that χ is an isomorphism of Poisson manifolds will be called a Poisson manifold of solutions of Euler–Lagrange equations. The following theorem deals with the relation between Poisson manifolds of solutions of Euler–Lagrange equations and Poisson manifolds of geodesic arcs.

Theorem 2. Let L_{π} : W (ℝ × X) → W (ℝ) be a fibred Poisson manifold of geodesic arcs such that all fibers are Poisson submanifolds, fibers over constant arcs are Abelian and fibers over non-constant arcs are symplectic. Then W (ℝ×X) is a union of three disjoint Poisson submanifolds such that the two of them are isomorphic to certain Poisson manifold of solutions of Euler–Lagrange equations and the third of them is an Abelian fibration of manifolds of geodesic arcs of Levi–Civita connection on X.

Proof. Let us consider a chart x ∈ At(ℝ × X) such that x^{0} is a coordinate on ℝ, x^{1}, x^{2}, ..., x^{dim X} are coordinates on X and rng x is an open ball in ℝ^{dim X+1}. Since W (ℝ × X) is a fibred manifold of geodesic arcs over basis W (ℝ), we get R_{jkl}^{i} = 0 for i = 0. Hence, the coordinate x^{0} may be chosen in such a way that for i = 0

(21)  Γ_{jk}^{i} = 0.

Since the fibers over constant arcs are Abelian, by (10) and (11) we obtain h^{ij} = 0, g_{k}^{ij} = 0, where k ≠ 0. Since the fibers over non-constant arcs are Poisson submanifolds, by (11) we obtain g_{0}^{i0} = g_{0}^{0i} = 0.

Throughout all the following text we will not consider the zero value of indices i, j, k, l, m. We denote g_{0}^{ij} = g^{ij} and x^{0} = t. Taking into account that the fibers over non-constant arcs are symplectic we obtain

(22)  det(g^{ij}) ≠ 0

at each point of X. By (6) g^{ij} = g^{ji} follows. Relations (3) - (5) are fulfilled identically. Relation (7) can be expressed in the form

(23)  \frac{∂g^{ij}}{∂x^{k}} + Γ_{km}^{i}g^{mj} + Γ_{km}^{j}g^{im} = 0, \frac{∂g^{ij}}{∂t} + Γ_{0m}^{i}g^{mj} + Γ_{0m}^{j}g^{im} = 0.

By a straightforward computation we get

Γ_{jk}^{i} = \frac{1}{2}g^{il}(\frac{∂g_{lj}}{∂x^{k}} + \frac{∂g_{kl}}{∂x^{j}} − \frac{∂g_{jk}}{∂x^{l}} ), (24)

Γ_{j0}^{i} = Γ_{0j}^{i} = \frac{1}{2}g^{il}(\frac{∂g_{lj}}{∂t} + Φ_{lj}), (25)

where g_{ij} are solutions of equations g_{ij}g^{jk} = δ_{i}^{k} and Φ_{ij} are functions on dom x such that Φ_{ij} + Φ_{ji} = 0. Relation (8) can be written according to (24) and (25) in the form

(26)  \frac{∂Φ_{ij}}{∂x^{k}} + \frac{∂Φ_{jk}}{∂x^{i}} + \frac{∂Φ_{ki}}{∂x^{j}} = 0, \frac{∂Φ_{ij}}{∂t} + \frac{∂F_{j}}{∂x^{i}} − \frac{∂F_{i}}{∂x^{j}} = 0,

where F_{i} = g_{ij}Γ_{00}^{j}. Thence the Poincaré lemma enables us to show that there exist functions A_{i}, φ on dom x satisfying

(27)  Φ_{ij} = \frac{∂A_{i}}{∂x^{j}} − \frac{∂A_{j}}{∂x^{i}} , F_{i} = \frac{∂A_{i}}{∂t} − \frac{∂φ}{∂x^{i}} .

Since the Lagrange function

(28)  L = \frac{1}{2}g_{ij}ẋ^{i}ẋ^{j} + A_{i}ẋ^{i} + φ

exists on J^{1}(dom x), the components

(29)  ϵ_{i} = g_{ij}(ẍ^{j} + Γ_{kl}^{j}ẋ^{k}ẋ^{l} + 2Γ_{k0}^{j}ẋ^{k} + Γ_{00}^{j})

of a π_{20}-horizontal 1-contact differential 2-form on J^{2}(ℝ × X) are Euler–Lagrange expressions. According to (18) and (22) they are regular. Let us denote the corresponding manifold of solutions of Euler–Lagrange equations by E(ℝ × X).

In view of (21) W (ℝ) = W_{1}(ℝ) ∪ W_{2}(ℝ) ∪ W_{3}(ℝ), where W_{1}(ℝ) is the open submanifold of all increasing geodesic arcs, W_{2}(ℝ) is the open submanifold of all decreasing geodesic arcs and W_{3}(ℝ) is the closed submanifold of all constant geodesic arcs. Let us decompose the manifold W (ℝ × X) into the three disjoint submanifolds W_{1}(ℝ × X) = L_{π}^{−1}(W_{1}(ℝ)), W_{2}(ℝ × X) = L_{π}^{−1}(W_{2}(ℝ)) and W_{3}(ℝ × X) = L_{π}^{−1}(W_{3}(ℝ)). These submanifolds are Poisson ones because of W_{1}(ℝ × X), W_{2}(ℝ × X) are open ones and W_{3}(ℝ × X) is by assumption the union of Abelian Poisson manifolds over constant geodesic arcs. Clearly, the submanifold W_{3}(ℝ×X) is Abelian too.

Let us suppose γ ∈ W_{1}(ℝ×X). Then there exists a unique mapping σ such that dom σ = rng(π ∘ γ), γ = σ ∘ π ∘ γ. It holds σ ∈ E(ℝ × X). The induced mapping W_{1}(ℝ × X) → E(ℝ × X) is an isomorphism of Poisson manifolds.

Let us suppose γ ∈ W_{2}(ℝ×X) and define the mapping μ : ⟨0, 1⟩ ∋ τ → (1−τ) ∈ ⟨0, 1⟩. In such a case γ ∘ μ = R_{μ}(γ) ∈ W (ℝ × X). Since π ∘ γ and μ are decreasing mappings, π ∘ γ ∘ μ is an increasing mapping and so γ ∘ μ ∈ W_{1}(ℝ × X). Since R_{μ} is a Poisson mapping and since R_{μ}^{−1} = R_{μ} holds, the mapping W_{2}(ℝ × X) ∋ γ → γ ∘ μ ∈ W_{1}(ℝ × X) is an isomorphism of Poisson manifolds. Consequently, the fact that there is an isomorphism between W_{1}(ℝ × X) and E(ℝ × X) implies the existence of isomorphism between W_{2}(ℝ × X) and E(ℝ × X).

For each a ∈ ℝ the set π^{−1}(a) is a geodesic submanifold of the manifold ℝ × X that is diffeomorphic to X. According to (24) the connection on π^{−1}(a) is the Levi– Civita one. Let W (π^{−1}(a)) be the manifold of geodesic arcs of this connection. Taking into account

(30)  W_{3}(ℝ × X) = ⋃_{a∈ℝ} W (π^{−1}(a)),

we see that W_{3}(ℝ × X) is the Abelian fibration of manifolds of geodesic arcs of Levi–Civita connection on X over basis ℝ. This completes the proof.

References

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3. S. Kobayashi and K. Nomizu, Foundations of differential geometry, Volume I (Interscience Publishers, New York, London, 1963), 113–153.

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