Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 265–271

NATURAL CONNECTIONS OF 1–FORMS ON TANGENT BUNDLES

Anton Dekrét^{1}

Abstract. Ideas of the Lagrange formalism are used to constructions of connections from a scalar 1-form on T M. Keywords. Semibasic scalar and vector forms, semispray, connections, symplectic forms. MS classification. 58F05, 70H35.

In the Lagrange formalism of classical mechanics connections are a useful tool, see for example [1], [2], [6], [7], [8]. Let us recall that a time independent lagrangian L : T M → R of first order the mechanical system is a semispray S_{L} on T M such that the basic equation i_{S_{L}}dω = dE is satisfied, where ω = J^{∗}dL, E = L − CL, J is the canonical morphism on T M and C is the Liouville field on T M, see Godbillon [4]. Griffone, [5], has shown that if S is a semispray on T M then the Lie derivative L_{S}J is a (1, 1)-form on T M determining a connection Γ_{S} on T M. In [3] we have proved that every connection constructed by natural way of order one from a semispray S is of the form Γ_{S}+cJ, c ∈ R. In this paper we show that the set of connection connected with every regular lagrangian L is larger that the one connected with S_{L}. We deal with this problem in a little more general case of 1-forms on T M.

Let M be a smooth manifold, m = dim M. Let (x^{i}) be a local chart on M. Denote (x^{i}, x_{1}^{i}) the induced chart on the tangent bundle p_{M} : T M → M. In this chart the Liouville field C or the canonical morphism is of the form C = x_{1}^{i}∂/∂x_{1}^{i} or J = dx^{i} ⊗ ∂/∂x_{1}^{i}. Recall that a semispray is a vector field S on T M such that J(S) = C and that a spray is semispray such that [C, S] = S.

In this paper a connection Γ on T M is understood in a some of the following equivalent senses:

1. Γ is a global section T M → JT M of the first jet prolongation of T M.

2. Γ is a distribution of Γ-horizontal subspaces HΓ ⊂ T T M such that T T M = V T M ⊕ HΓ, where V T M → T M is the vector bundle of all vertical vectors on T M.

3. Γ is given by the Γ-horizontal projection h_{Γ} that is a vector 1-form on T M such that h_{Γ}(Y) = 0 for any Y ∈ V T M and T p_{M} ·h_{Γ} = T p_{M}, i.e. h_{Γ} = dx^{i}⊗∂/∂x^{i}+Γ_{j}^{i}dx^{j}⊗∂/∂x_{1}^{i} in the induced chart.

4. Γ is determined by the Griffone form on T M that is a vector 1-form φ on T M such that φJ = −J, Jφ = J, i.e.,

φ = dx^{i} ⊗ ∂/∂x^{i} + (φ_{j}^{i}dx^{j} − dx_{1}^{i}) ⊗ ∂/∂x_{1}^{i}.

Typeset by AMS-TEX

1This paper is in final form and no version of it will be submitted for publication elsewhere.

The map φ ↦ \frac{1}{2}(φ+id_{T T M}) = h_{Γ} is a bijection between the set of the Griffone forms φ and the set of the horizontal projections h_{Γ}.

Let us recall that a scalar or a vector p-form ω on a fibred manifold π : Y → M is semibasic if ω(X_{1}, …, X_{p}) = 0 whenever at least one vector X_{i} is π-vertical. Then if h_{Γ} is a Γ-horizontal projection of a connection Γ and ψ is a semibasic V T M-valued 1-form on T M then h_{Γ} = ψ is the horizontal projection of another connection on T M which will be denoted by Γ + ψ.

Properties of semibasic forms on T M

Let γ = γ_{i}(x, x_{1})dx^{i} be a semibasic 1-form on T M. It determines the morphism γ̄ : T M → T^{∗}M, γ(X)(Z) = γ(Z_{1}), T p_{M}(Z_{1}) = Z, Z_{1} ∈ T_{X}T M, i.e. z_{i} = γ_{i}(x, x_{1}) in the induced chart (x^{i}, z_{i}) on T^{∗}M, see [4].

Denote V γ̄ : T γ̄ |_{V T M}: V T M → V T^{∗}M the restriction of the tangent prolongation of γ̄ to V T M Let p_{2} : V T^{∗}M ≡ T^{∗}M ×_{M} T^{∗}M → T^{∗}M be the projection on the second factor. Using the identification V^{∗}T M := p_{M}^{∗}T^{∗}M ≡ T M ×_{M} T^{∗}M we get the map V γ : V T M → V^{∗}T M defined by the pull-back p_{M}^{∗}(p_{2} · V γ̄), i.e. V γ is a section T M → V^{∗}T M ⊗ V^{∗}T M, V γ = γ_{ij_{1}}dx_{1}^{i} ⊗ dx_{1}^{j}, where we use the notations \frac{∂f}{∂x^{i}} := f_{i}, \frac{∂f}{∂x_{1}^{i}} := f_{i_{1}}.

A semibasic form γ is said to be V -regular (V -symmetric) if V γ is a regular (symmetric) bilinear form. In the case of a V -regular form, V γ̃ = γ̃^{ij}∂/∂x_{1}^{i}⊗∂/∂x_{1}^{j}, γ̃^{ij}γ_{jk_{1}} = ∂_{k}^{i} denotes the inverse map of V γ. Obviously γ is V -regular iff γ̄ is regular.

Recall that a vector field X = ξ^{i}∂/∂x^{i} + η^{i}∂/∂x_{1}^{i} on T M is projectable if T p_{M}X = ξ^{i}∂/∂x^{i} is a vector field on M. The field T X = ξ^{i}∂/∂x^{i} + \frac{∂ξ^{i}}{∂x^{j}} x_{1}^{j}∂/∂x_{1}^{i} the flow of which is the tangent prolongation of the flow of a vector field X = ξ^{i}∂/∂x^{i} on M is projectable.

A field X : T M → T T M on T M determines the map T p_{M} · X : T M → T M, x^{−i} = x^{i}, x_{1}^{−i} = ξ^{i}(x, x_{1}). Then the map V (T p_{M} · X) : V T M → V T M is a vector bundle morphism over T p_{M} · X. Using the projection p_{2} : V T M ≡ T M ×_{M} T M → T M on the second factor we get the map

V X := p_{2} · V (T p_{M} · X) : V T M → T M, x^{i} = x^{i}, x_{1}^{i} \frac{∂ξ^{i}}{∂x_{l}^{j}} dx_{l}^{j}

that is a section T M → T^{∗}M ⊗_{T M} V T M ≡ V^{∗}T M ⊗ V T M ≡ T^{∗}M ⊗_{T M} T M. We say that a vector field X is a b-regular if V X is regular, (locally if det ξ_{j_{1}}^{i} ≠ 0). Every semispray on T M is regular.

Recall that a semibasic 1-form γ is closed iff γ = p_{M}^{∗}φ, where φ is a closed 1-form on M. The Lie-derivative L_{X}γ of a semibasic 1-form γ with respect to a vector field X on T M is of the coordinate form

(1)  L_{X}γ = (γ_{ij}ξ^{j} + γ_{j}ξ_{i}^{i} + γ_{ij_{1}}η^{j})dx^{i} + γ_{j}ξ_{i_{1}}^{j}dx_{1}^{i}.

Denote by V_{γ} or V_{X} the space of all vector fields or all semibasic forms on T M such that L_{X}γ is semibasic. Any projectable vector field belongs to V_{γ}. A semibasic form γ belongs to V_{X} iff γ ∈ Ker(V X)^{∗} where (V X)^{∗} : V^{∗}T M → V^{∗}T M is the join morphism to V X. It is clear that if X is b-regular then V_{X} = {0}. From (1) it follows Lemma 1. Let a semibasic form γ is V -regular. Then for every X ∈ V γ and for any semibasic form ε there exists a unique field X̄ on T M such that T p_{M}X = T p_{M} X̄ and Lx̄γ = ε.

Let Y = c^{i}∂/∂x^{i} be a vector field on M. Let T^{∗}Y = c^{i}∂/∂x^{i} − \frac{∂c^{k}}{∂x^{i}} z_{k}∂/∂z_{i} denote the vector field on T^{∗}M induced by the flow (T ϕ)^{∗}, where ϕ is the flow of the field Y.

Lemma 2. Let X be a projectable vector field on T M over a vector field Y on M. Then the vector field X̄ described in Lemma 1 for ε = 0 satisfies T γ̄(X) = T^{∗}Y.

Proof. If Y = c^{i}∂/∂x^{i} then X̄ = c^{i}∂/∂x^{i} + η^{i}∂/∂x_{1}^{i} where η^{i} = γ̃ is (−γ_{sj}c^{j} − c_{s}^{j}γ_{j}). Then T γ̄(X̄) = c^{i}∂/∂x^{i} + (γ_{ij}c^{j} − γ_{ij}c^{j} − c_{i}^{j}γ_{j})∂/∂z_{i} = T^{∗}Y. □

If γ is a V -regular then γ̄ : T M → T^{∗}M is an isomorphism of the symplectic structures (T M, dγ) and (T^{∗}M, dλ), where λ = z_{i}dx^{i} is the Liouville form on T^{∗}M. Let I : T (T M) → T^{∗}M denote the map defined by the contraction I(X) = i_{X}dγ = α_{i}dx^{i} + β_{i}dx_{1}^{i} where

(2)  α_{i} = (γ_{ij} − γ_{ji})ξ^{j} + γ_{ij_{1}}η^{j}, β_{j} = −γ_{ij_{1}}ξ^{i}.

Recall that a vector field X on T M is a dynamical system of the structure (T M, dγ) if the form I(X) = i_{X}dγ is closed. Since the Lie derivative L_{X}γ = (i_{x}d = di_{X})^{γ} then X is a dynamical system of (T M, dγ) iff L_{x}γ is closed or iff L_{X}dγ = 0. If X is a symmetry of γ, (L_{X}γ = 0), then is a dynamical system. Therefore by Lemma 1 to any vector field X ∈ V_{γ} there is a unique dynamical system X̄ of (T M, dγ) such that LX̄γ = 0.

A vector field X on T M is called to be a quasidynamical system of (T M, dγ) if J^{∗}(dL_{X}γ) = 0. Every projectable vector field on T M is a quasidynamical systems of (T M, dγ).

Lemma 3. A semispray on T M is the quasidynamical system of (T M, dγ) iff γ is V -symmetric.

Proof follows from J^{∗}dL_{X}γ = γ_{it_{1}}ξ_{k_{1}}^{i}dx^{t} ∧ dx^{k}, ξ_{k_{1}}^{i} = δ_{k}^{i}.

According to (2) it is clear that if γ is V -regular then I is a bijection from the set of all vertical vectors onto the set of all semibasic forms on T M. Therefore the Poisson bracket (α, β) of semibasic forms α, β is semibasic.

Lemma 4. The Poisson bracket (α, β) of two semibasic forms α, β satisfies (α, β) = ⟨V β ⊗ α − V α ⊗ β, Ṽ ε⟩ where the bracket ⟨ ⟩ denotes the contraction.

Corollary. Let α = p_{M}^{∗}α_{0}, β = p_{M}^{∗}β_{0} be the p_{M}-pullback of forms α_{0}, β_{0} on M. Then (α, β) = 0.

It is easy to show that the vector field [C, X] is vertical iff C ∈ Ker V X. If X_{1}, X_{2} are semispray on T M then [X, X̄] is a semispray iff X_{2} = X_{1} − C.

Scalar 1-forms and connections on T M

Let ε = a_{i}dx^{i} + b_{i}dx_{1}^{i} be a scalar 1-form on T M. Then γ = J^{∗}ε = ε · J = b_{i}dx^{i} is semibasic on T M. We say that ε is V -symmetric or V -regular if γ = J^{∗}ε is V - symmetric or V -regular respectively. For the sections V J^{∗}ε : T M → V^{∗}T M × V^{∗}T M { \atop V J^{∗}ε} : T M → V T M ⊗ V T M we will use the shortened notations V ε, Ṽ ε.

We are interested in the constructions of connections on T M from the form ε. First we will use the Lagrange formalism in the classical mechanics.

It is clear that the form dγ = dJ^{∗}ε = b_{ij}dx^{j} ∧ dx^{i} + b_{ij_{1}}dx_{1}^{i} ∧ dx^{i} is symplectic iff ε is V -regular. Denote A := ε(C) = b_{s}x_{1}^{s}.

Let λ = λ_{i}(x, x_{1})dx^{i} + λ̄(x, x_{1})dx_{1}^{i} be a 1-form on T M.

Lemma 5. If ε is a V -regular scalar 1-form on T M then the vector field X on T M satisfying the equation

(3)  i_{X}dJ^{∗}ε = ε − dA + λ

is a semispray if and only if λ is semibasic.

Proof follows from the coordinate form

(b_{ij} − b_{ji})ξ^{j} + b_{ij_{1}}η^{j} = a_{i} − b_{si}x_{1}^{s} + λ_{i}, b_{sj−1}ξ^{s} = b_{sj_{1}}x_{1}^{s} − λ̄_{j}

of the equation (3).

We will suppose that λ is semibasic. The semispray X from the equation (3) will be denoted by X = S_{ε}λ and called the Lagrange field of the structure (ε, λ). In the case λ = 0 we use the denotation S_{ε} and we say that S_{ε} is the Lagrange field of the V -regular 1-form ε.

Remark. If ε is semibasic and V -regular then dε is symplectic. We can use A = ε(S) = b_{s}x_{1}^{s}, where S is an arbitrary semispray but the solution X of the equation i_{X}dε = ε−dA is not a semispray.

We will suppose that ε is not semibasic and is V -regular. Then the Lie derivatives L_{S_{ελ}}J, L_{S_{ε}}J are the Griffone forms and determine the connections Γ_{ελ}, Γ_{ε} the horizontal projection of which are of the coordinate form

h_{Γ} = dx^{i} ⊗ ∂/∂^{i} + Γ_{j}^{i}dx^{j} ⊗ ∂ ∂x_{1}^{i}, Γ_{j}^{i} = \frac{1}{2} \frac{∂B^{i}}{∂x_{1}^{j}}

where B^{i} are the coordinates of the semisprays S_{ελ} or S_{ε}.

It is clear that the connections Γ_{ελ} + cJ, c ∈ R, are not the only connections determined by the forms ε and λ. For example Γ_{S_{ελ}} + kJ^{∗}ε ⊗ C + f(L, V ε(C, C))J, where k ∈ R and f is a smooth function of two variables, are another connections on T M. We will construct some geometrical objects, that provide to construct large families of connections from ε.

a) We will prove that if ε is a V -symmetric 1-form on T M then (a_{ij_{1}} − b_{ji}) are the coordinates of a tensor field β_{ε} : T M → V^{∗}T M ⊗ V^{∗}T M ≡ T^{∗}M ⊗_{T M} T^{∗}M. Let (x^{i}, x_{10}^{i}, x_{01}^{i}, x_{11}^{i}), (x^{i}, x_{100}^{i}, x_{010}^{i}, x_{110}^{i}, x_{001}^{i}, x_{101}^{i}, x_{011}^{i}, x_{111}^{i}) be the induced charts on T T M and T T T M. Let us recall that the canonical involution i_{3} on T T T M ≡ T_{2}(T M) is of the coordinate form

i_{3}(x^{i}, x_{100}^{i}, …, x_{111}^{i}) = (x^{i}, x_{100}^{i}, x_{001}^{i}, x_{101}^{i}, x_{010}^{i}, x_{110}^{i}, x_{011}^{i}, x_{111}^{i}).

Consider the form ε as a function ε = a_{i}x_{01}^{i} +b_{i}x_{11}^{i} on T T M. Let p_{2} : T R ≡ R ×R → R is the projection on the second factor. Then

p_{2}T ε − p_{2} · T ε · i_{3} = (a_{ij} − a_{ji})x_{010}^{i}x_{001}^{i} + (a_{ij_{1}} − b_{ji})x_{010}^{i}x_{101}^{j} + (b_{ij} − a_{ji})x_{110}^{i}x_{001}^{j} is a function on T T T M and its restriction to V (T p_{M}) is of the coordinate form ε̄ = (a_{ij_{1}} −b_{ji})x_{010}^{i}x_{101}^{i} where we use the denotation V (π) for the space of all vertical vectors of a surjective submersion π : Y → M. Let X = c^{i}∂/∂x^{i}, Y = y^{i} = ∂/∂x^{i} ∈ T_{x}Y. There exists Z ∈ V_{h}(T p_{M}), h ∈ T_{x}M such that T p_{M} · p_{T T M}(Z) = X · p̄_{2} · T p_{T M}(Z) = Y, where p̄_{2} : V T M ≡ T M ×_{M} T M → T M is the projection on the second factor. For example, Z = (x^{i}, x_{100}^{i}, c^{i}, x_{100}^{i}, 0, y^{i}, 0, x_{111}^{i}). Put β_{ε}(X, Y) = ε̄(Z) = (a_{ij_{1}} − b_{ji})c^{i}y^{j}. Obviously β_{ε} : T M → T^{∗}M ⊗_{T M} T^{∗}M.

b) We will show that if α : T M → V T M ⊗ V T M, (x^{i}, x_{1}^{i}) ↦ (x^{i}, x_{1}^{i}, x^{ij} = α^{ij}) is a section then the functions α_{k_{1}}^{ij} = ∂α^{ij}/∂x_{1}^{k} determine a section V̄ α : T M → T^{∗}M ⊗ (V T M ⊗ V T M). Denote κ_{1} : V T M ⊗ V T M → T M, κ_{2} = p_{M} · κ_{1} the canonical submersions. Using the identification (V T M⊗_{T M} V T M)_{x} ≡ T_{x}M×(T_{x}M⊗T_{x}M) we get the other identification T (V T M ⊗ V T M)_{x} ≡ T (T_{x}M) × T (T_{x}M ⊗ T_{x}M) = (V T M)_{x} × T (T_{x}M ⊗ T_{x}M). Then we can extend the map α onto ξ_{α} : V T M → T (V T M ⊗ V T M), (x^{i}, x_{1}^{i}, 0, dx_{1}^{i}) ↦ (x^{i}, x_{1}^{i}, α^{ij}, 0, dx_{1}^{i}, 0). Using the map V α = T α |_{V T M} we obtain the map V̄ α := V α − ξ_{α} : (x^{i}, x_{1}^{i}, 0, dx_{1}^{i}) ↦ (x^{i}, x_{1}^{i}, α^{ij}, 0, 0, α_{k_{1}}^{ij}dx_{1}^{k}) = (x^{i}, x_{1}^{i}, α_{k_{1}}^{ij}dx_{1}^{k}) which defines the section \overline{V α} : T M → T^{∗}M ⊗ (V T M ⊗ V T M).

c) Remember that [C, S] is a semispray and then S^{∗} = [C, S] − S_{ε} is a vertical vector field. In the case of a V -regular and V -nonsymmetric or of a V -regular and V - symmetric form ε denote by K the set K = \{C, S_{ε}, ε, ω = J^{∗}ε, V ε, α = Ṽ ε, \overline{V α}, J\} or K = \{C, S_{ε}, J^{∗}ε, V ε, α = Ṽ ε, \overline{V α}, β_{ε}, J\}. Using tensor products and contractions of objects of K we can construct a family of semibasic V T M-value 1-forms φ on T M and the connections Γ_{ελ} + φ joined to ε.

Let us construct connections for which the field S_{ε} is horizontal. Let Γ be an arbitrary connection on T M, h_{Γ} = dx^{i} ⊗ ∂/∂x^{i} + Γ_{j}^{i}dx^{j} ⊗ ∂/∂x_{1}^{i}. Then there exists a unique semispray S_{Γ} which is Γ-horizontal, S_{Γ} = h_{Γ}(S) = x_{1}^{i}∂/∂x^{i} + Γ_{j}^{i}x_{1}^{j}∂/∂x_{1}^{i}, where S is an arbitrary semispray on T M. We say that S_{Γ} is associated to Γ. The values of the semisprays S_{Γ} on the zero-section 0 : M → T M, 0(x) = 0, are equaled to zero-vectors.

Let S_{ελ} be the Liouville field of (ε, λ). The necessary condition for S_{ελ} to be globally associated to Γ is: S_{ελ}(0) = 0. Every connection Γ on T M can be expressed in the form Γ = Γ_{ε} + φ, where φ is a semibasic 1-form on T M with values in V T M. Let S denote the semispray associated to the connection Γ_{ε} + φ. We have

S_{φ} = x_{1}^{i}∂/∂x^{i} + (Γ_{j}^{i} + φ_{j}^{i})x_{1}^{j}∂/∂x_{1}^{i}, Γ_{j}^{i} = \frac{1}{2}B_{j_{1}}^{i}, B^{i} = b̃^{is}(a_{s} − b_{sk}x_{1}^{k}).
S_{ελ} = x^{i}∂/∂x^{i} + b̃^{is}(a_{s} − b_{sk}x_{1}^{k} + λ_{s})∂/∂x_{1}^{i} = x_{1}^{i}∂/∂x^{i} + (B^{i} + b̃^{is}λ_{s})∂/∂x_{1}^{i}.

Then S_{ελ} = S_{φ} if and only if

λ_{i} = b_{it_{1}}(Γ_{j}^{t} + φ_{j}^{t})x_{1}^{j} − b_{it_{1}}B^{t}.

For any semispray S it holds φ(S) = φ_{t}^{i}x_{1}^{t}. Since S_{φ}−S_{ε} = [(Γ_{j}^{i} +φ_{j}^{i})x_{1}^{j}−B^{i}]∂/∂x_{1}^{i} then S_{ελ} = S_{φ} if and only if λ = ⟨V ε ⊗ (S_{φ} − S_{ε})⟩_{2} ≡ V ε(·, S_{φ} − S_{ε}). Also S_{φ} = S_{Γ_{ε}} + φ(S) and then S_{φ} − S_{ε} = S_{Γ_{ε}} − S_{ε} + φ(S), i.e. λ = ⟨V ε, S_{Γ_{ε}} − S_{ε} + φ(S)⟩_{2}. We have proved

Proposition 1. Let ε be a V -regular 1-form on T M. Let Γ = Γ_{ε} + φ be an arbitrary connection on T M. Then there is a unique semibasic 1-form λ = V ε(·, S_{Γ_{ε}} − S_{ε} + φ(S)) such that S_{φ} = S_{ελ}, i.e. such that the semispray associated to Γ_{ε} + φ is the Lagrange field of the structure (ε, λ). Conversaly, if λ is a given semibasic 1-form on T M then the semibasic (1, 1)-form φ with values in V T M (determining the connection Γ = Γ_{ε} + φ) such that S_{φ} = S_{ελ} satisfies φ(S) = S_{ε}−S_{Γ_{ε}}+Ṽ ε(·, λ), where S is an arbitrary semispray on T M.

Corollary. If φ is a semibasic (1, 1)-form with value in V T M such that S_{φ} = S_{ε} then φ(S) = S_{ε} − S_{Γ_{ε}} = (B^{i} − \frac{1}{2}B_{j_{1}}^{i}x_{1}^{j})∂/∂_{1}^{i}.

We will find semibasic (1, 1)-forms with values in V T M such that S_{φ} = S_{ε}. Consider, (compare with [1]),

S_{ε}^{∗} := [C, S_{ε}] − S_{ε} = (B_{s_{1}}^{j}x_{1}^{s} − 2B^{j})∂/∂x_{1}^{j}.

Then S_{ε} − S_{Γ_{ε}} = −\frac{1}{2}S_{ε}^{∗}. Let i_{c}^{1}V ε be a 1-form such that i_{c}^{1}V ε(Y) = V ε(C, Y). Then we get the section i_{c}^{1}V ε ⊗ S_{ε}^{∗} : T M → V^{∗}T M ⊗ V T M. Put

φ_{0} = −\frac{1}{2} \frac{1}{V ε(C, C)}i_{c}^{1}V ε ⊗ S_{ε}^{∗} = \frac{1}{b_{kt_{1}}x_{1}^{k}x_{1}^{t}} [B^{i} − \frac{1}{2} \frac{∂B^{i}}{∂x_{i}^{j}} x_{1}^{j}]b_{kt_{1}}x_{1}^{k}dx^{t} ⊗ ∂/∂x_{1}^{i},

i.e. φ_{0} is a semibasic (1, 1)-form with values in V T M defined on T M − 0^{V ε}, where 0^{V ε} denotes the set of all vectors in T M such that V ε(C, C) = 0. When V ε is positively definite then 0^{V ε} is the set of all zero-vectors on M. Obviously φ_{0}(S) = S_{ε} − S_{Γ_{ε}}.

Proposition 2. The Lagrange field S_{ε} of a V -regular form ε is the semispray associated to the connection Γ_{ε} + φ_{0}.

Proof. If S_{φ_{0}} is associated to the connection Γ_{ε} + φ_{0} then

S_{φ_{0}} = x_{l}^{i}∂/∂x^{i} + [\frac{1}{2}B_{j_{1}}^{i}x_{1}^{j} + \frac{1}{b_{kt_{1}}x_{1}^{k}x_{1}^{t}} (B^{i} − \frac{1}{2}B_{j_{1}}^{i}x_{1}^{j})b_{kt_{1}}x_{1}^{k}x_{1}^{t}]∂/∂x_{1}^{i} = S.

□

Remark 1. If S_{ε} is a spray then S_{ε}^{∗} = 0 and therefore S_{ε} = S_{Γ_{ε}}. That is the well known result that if S is a spray then S is associated to Γ_{ε}.

Remark 2. Put

φ_{1} = −\frac{1}{2} \frac{1}{ε(C)}J_{1}^{∗}ε ⊗ S_{ε}^{∗}, J^{∗}ε = b_{i}dx^{i},

i.e. φ_{1} : T M → T^{∗}M ⊗ V T M is a section defined for ε(C) ≠ 0. As J^{∗}ε(S) = ε(C) for any semispray S therefore φ_{1}(S) = S_{ε} − S_{Γ_{ε}} if ε(C) ≠ 0. Then the connection Γ_{ε} + φ_{1} is such that S_{ε} is associated to Γ_{ε} + φ_{1}.

Remark 3. Denote ε^{V} : ε |_{V T M}. Put α = Ṽ ε, h = \overline{V α}(ε^{V} , ε^{V} , C) = b_{k}^{ij}b_{i}b_{j}x_{1}^{k}, β = ⟨\overline{V α} ⊗ (ε^{V} ⊗ ε^{V})⟩ = b̃_{k}^{ij}b_{i}b_{j}dx^{k}. Obviously β(S) = h. Then the semibasic (1, 1)-form φ_{2} = −\frac{1}{2} \frac{1}{h}β ⊗ S_{ε}^{∗} is such that S_{ε} is associated to the connection Γ_{ε} + φ_{2} for h ≠ 0.

Remark 4. If L : T M → R is a lagrangian then ε = dL, (a_{i} = L_{i}, b_{i} = L_{i_{1}}) is V - symmetric and it is V -regular iff lagrangian is regular. We use denotation S_{L} ≡ S_{ε}, Γ_{L} = Γ_{ε}. Since a_{ij_{1}} = b_{ji} therefore β_{ε} = 0. Now the form φ_{0} coincides with the form considered in [1] where the connection Γ_{L} + φ_{0} is called canonical. We see that it is not a unique connection on T M such that S_{L} is associated with it. There are another connections of this property which domains are in general open submanifolds of T M.

If d(J^{∗}ε) is symplectic then every connection Γ on T M determines the connection Γ̄ = Orth Γ which is orthogonal to Γ according to dω, i.e. dω(X, Y) = 0 for any Γhorizontal vector X and any Γ̄-horizontal vector Y. If Γ_{j}^{i} are the coordinate functions of Γ then the functions of Γ̄ are of the form Γ̄_{k}^{i} = b̃^{is}(b_{ks} − b_{sk} + b_{ku_{1}}Γ_{s}^{u}). A connection Γ is called the Lagrange connection if Γ̄ = Γ. It is easy to deduce that if ε is V -symmetric and V -regular then the connection Γ = Γ_{ελ} is the Lagrange connection if and only if a_{kp_{1}} − b_{pk} − a_{pk_{1}} + b_{kp} = λ_{pk_{1}} − λ_{kp_{1}}. It gives Proposition 3. Let ε be V -symmetric and V -regular. Then the connection Γ_{ελ} is the Lagrange connection if a(β_{ε}) = −a(V λ) where a denotes the antisymmetrization of forms.

Corollary. If ε = dL then the connection Γ_{ελ} is the Lagrange connection if and only if λ is V -symmetric.

Remark. We see that there is a large family of connections on T M constructed by the connection Γ_{ελ} and the set K. The mapping Γ ↦ Γ̄ = Orth Γ gives another connections. Using the procedure of the theory of natural operators it is possible to solve the problem of the constructions of all connections deduced from a V -regular scalar 1-form ε on T M. But it is not such interesting as the question of deeper informations about the geometry of ε (and thus of the regular lagrangian) obtained by connections constructed from ε.

References

1. L.C. de Andrés, M. de León and P.R. Rodrigues, Canonical connections associated to regular lagrangians of higher order, to appear.

2. M. Crampin, Alternative lagrangians in particle dynamics, Diff. Geom. and its Applications, Proc. Conf., 1986, Brno, 1–12.

3. A. Dekrét, Mechanical structures and connections, Diff. Geometry and Applications, Proc. Conf., 1988, Dubrovnik, 121–132.

4. C. Godbillon, Geometrie differentielle et mécanique analytique, Paris, 1969.

5. J. Griffone, Structure presque-tangent et connections I, Ann. Inst. Fourier (Grenoble) 22 (1972), 287–334.

6. J. Klein, Almost symplectic structures in dynamics, Diff. Geom. and its Applications, Proc. Conf., 1986 Brno, 79–90.

7. M. de León and P.R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland, 1985, 290 p.

8. A. Vondra, Semisprays, connections and regular equations in higher order mechanics, Diff. Geom. and its Applications, Proc. Conf., Brno 1989, 276–287.

Anton Dekrét TU, Masarykova 24 960 53 Zvolen Slovak Republic

Received 27 August 1992