On the Geometries of Superior Order

Radu Miron

The geometries of higher order, defined in the present paper as the study of the category of jet bundles (J_{o}^{k}M, π, M), were suggested by the old problem of the prolongations to J_{o}^{k}M of Riemannian structures g apriori given on the base manifold M. In the case k = 1 there are very good examples: the geometry of Finsler spaces and the geometry of Lagrange spaces, [3].

For k > 1 there are some geometrical results concerning the analytical Mechanics of superior order and there are some important studies on the total space of jet bundles. Recently, together with Gh.Atanasiu, we have considered the notion of k-osculator bundle (Osc^{k}M, π, M), which can be identified to the bundle (J_{0}^{k}M, π, M) and, for it, we have studied:

a. The Geometry of the total space Osc^{k}M: vertical distributions, Liouville vector fields, k-tangent structure, sprays, nonlinear connections, N-linear connections, curvatures, torsions, structure equations etc.

b. Taking this oportunity we solved the problem of the prolongation to Osc^{k}M of the Riemannian structure g apriori given on M. The same problem for Finsler and Lagrange structures can be formulated.

c. We have defined the notion of higher order Lagrange spaces and studied for them the most important geometrical object fields.

d. Recently, to the Tsagas’ workshop, in December 1995, at the ”Aristoteles” University of Thessaloniki, the author presented the theory of subspaces in the higher Lagrange spaces.

In the present paper I shall describe, in synthesis, the construction of these geometries.

1. Let M be a real n-dimensional C^{∞}-manifold and (Osc^{k}M, π, M) its kosculator bundle. We briefly describe the manifold E = Osc^{k}M. Two curves in M, σ, ρ : I → M, with x_{0} ∈ M as common point, x_{0} = σ(0) = ρ(0), 0 ∈ I, have at x_{0} a ”contact of order k”, (k ∈ N^{⋆}), if for any function f ∈ F(U), x_{0} ∈ U, (U being an open set in M), satisfy

\frac{d^{α}}{dt^{α}} (f ∘ σ)(t) |_{t=0}= \frac{d^{α}}{dt^{α}} (f ∘ ρ)(t) |_{t=0}, (α = 1, ..., k).

The relation ”contact of order k” is a relation of equivalence. We denote by [ρ]_{x_{0}} a class of equivalence and named it a ”k-osculator space” in the point x_{0} to the manifold M.

Balkan Journal of Geometry and Its Applications, Vol.1, No.2, 1996, pp. 37-45 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Let Osc_{x_{0}}^{k} be the set of k-osculator spaces in x_{0} to M and let us consider the set

Osc^{k}M = ⋃_{x_{0}∈M} Osc_{x_{0}}^{k}

and the mapping

π : ([ρ]_{x_{0}}) ∈ Osc^{k}M ⟶ x_{0} ∈ M.

On the set Osc^{k}M there is a natural differentiable structure induced by that of M such that π becomes a differentiable submersion.

If (U, φ) is a local chart in M, x_{0} ∈ U and ρ : I → M is a curve represented by

x^{i} = x^{i}(t), t ∈ I, x_{0}^{i} = x^{i}(0), (0 ∈ I).

The class [ρ]_{x_{0}} has a representative element given by

x^{∗i}(t) = x^{i}(0) + t\frac{dx^{i}}{dt} (0) + ... + t^{k} \frac{1}{k!} \frac{d^{k}x^{i}}{dt^{k}} (0), t ∈ (−ε, ε) ⊂ I.

The previous polynomial functions are defined by the coefficients:

x_{0}^{i} = x^{i}(0), y_{0}^{(1)i} = \frac{dx^{i}}{dt} (0), ..., y_{0}^{(k)i} = \frac{1}{k!} \frac{d^{k}x^{i}}{dt^{k}} (0).

Therefore, the pair (π^{−1}(U), ϕ), with

ϕ : π^{−1}(U) ⊂ E ⟶ φ(U) × R^{kn}
ϕ([ρ]_{x_{0}}) = (x_{0}^{i}, y_{0}^{(1)i}, ..., y_{0}^{(k)i})

is a local chart on E, (i, j, h, ... = 1, ..., n)

So that, a differentiable atlas A_{M} on M determines a differentiable atlas A_{E} on E. The triple (Osc^{k}M, π, M) is a differentiable bundle.

The previous considerations show that the transformations of the local coordinates (x^{i}, y^{(1)i}, ..., y^{(k)i}) ⟶ (x̃^{i}, ỹ^{(1)i}, ..., ỹ^{(k)i}) have the form

(1.1)  \{ \begin{matrix}x̃^{i} & = x̃^{i}(x^{1}, ..., x^{n}), det ∥ \frac{∂x̃^{i}}{∂x^{j}} ∥≠ 0 \\ ỹ^{(1)i} & = \frac{∂x̃^{i}}{∂x^{j}} y^{(1)j} \\ .......... & ...................................................... \\ kỹ^{(k)i} & = \frac{∂ỹ^{(k−1)i}}{∂x^{j}} y^{(1)j} + ... + k \frac{∂ỹ^{(k−1)i}}{∂y^{(k−1)j}} y^{(k)j}, \end{matrix}

where we have

\frac{∂ỹ^{(α)i}}{∂x^{j}} = \frac{∂ỹ^{(α+1)i}}{∂y^{(1)j}} = ... = \frac{∂ỹ^{(k)i}}{∂ỹ^{(k−α)i}} , (α = 0, 1, ..., k − 1, y^{(0)} = x) etc.

2. Let us consider the natural basis

(\frac{∂}{∂x^{i}} , \frac{∂}{∂y^{(1)i}} , ..., \frac{∂}{∂y^{(k)i}} ),

of the tangent space T_{u}E, u = (x, y^{(1)}, ..., y^{(k)}) = (x^{i}, y^{(1)i}, ..., y^{(k)i}).

On E there exist k-vertical distributions V_{1}, ..., V_{k}. The distribution V_{1} is locally generated by the vector fields (\frac{∂}{∂y^{(1)i}} , ..., \frac{∂}{∂y^{(k)i}} ); V_{2} is a subdistribution

of V_{1}, locally generated by (\frac{∂}{∂y^{(2)i}} , ..., \frac{∂}{∂y^{(k)i}} ) etc ..., V_{k} is a subdistribution of

V_{k−1} locally generated by the vector fields (\frac{∂}{∂^{(k)i}} ).

All these distributions are integrable.

Also, on E there exist k-independent Liouville vector fields given by

Γ^{1}= y^{(1)i} \frac{∂}{∂y^{(k)i}} , Γ^{2}= y^{(1)i} \frac{∂}{∂y^{(k−1)i}} + 2y^{(2)i} \frac{∂}{∂y^{(k)i}} , ...,

(2.1)  Γ^{k}= y^{(1)i} \frac{∂}{∂y^{(1)i}} + ... + ky^{(k)i} \frac{∂}{∂y^{(k)i}} .

On the manifold E there exists a k-tangent structure, J : X (E) → X (E), such that

(2.2)  J(\frac{∂}{∂x^{i}} ) = \frac{∂}{∂y^{(1)i}} , J(\frac{∂}{∂y^{(1)i}} ) = \frac{∂}{∂y^{(2)i}} , ...,

J(\frac{∂}{∂y^{(k−1)i}} ) = \frac{∂}{∂y^{(k)i}} , J(\frac{∂}{∂y^{(k)i}} ) = 0.

Of course, we have J(Γ^{k}) = Γ^{k−1}, ..., J(Γ^{2}) =Γ^{1}, J(Γ^{1}) = 0 and J is an integrable structure.

A k-spray on E is a vector field S ∈ X (E) with the property

J(S) =Γ^{k} .

Then, S can be uniquely represented in the form:

(2.3)  S = y^{(1)i} \frac{∂}{∂x^{i}} + ... + ky^{(k)i} \frac{∂}{∂y^{(k−1)i}} − (k + 1)G^{i} \frac{∂}{∂y^{(k)i}} ,

G^{i}(x, y^{(1)}, ..., y^{(k)}) being the coefficients of the spray S.

A nonlinear connection on E = Osc^{k}M is a supplementary regular distribution N : u ∈ E → N_{u} ⊂ T_{u}E, to the vertical distribution V : u ∈ E → V_{u} ⊂ T_{u}E :

T_{u}E = N_{u} ⊕ V_{u}, ∀u ∈ E.

Setting N_{0} = N, N_{1} = J(N_{0}), ..., N_{k−1} = J(N_{k−2}) we get from the previous direct decomposition:

(2.4)  T_{u}E = N_{0}(u) ⊕ N_{1}(u) ⊕ ... ⊕ N_{k−1}(u) ⊕ V_{u}, ∀u ∈ E.

Consequently, to study the geometry of total space of the k-osculator bundle E = Osc^{k}M we shall use a fixed nonlinear connection N and we shall express the geometrical object fields on E by means of the direct decomposition (2.4). Also, N can be determined by some important geometric objects on E, like k-sprays or higher order Lagrangians.

A local basis adapted to (2.4) is given by

(2.5)  \{\frac{δ}{δx^{i}} , \frac{δ}{δy^{(1)i}} , ..., \frac{δ}{δy^{(k)i}} \}

where

(2.5)′  \frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − N_{(1)}^{j}_{i} \frac{∂}{∂y^{(1)j}} − ... − N_{(k)}^{j}_{i} \frac{∂}{∂y^{(k)j}}

and

\frac{δ}{δy^{(1)i}} = J (\frac{δ}{δx^{i}} ) , ..., \frac{δ}{δy^{(k)i}} = J (\frac{δ}{δy^{(k−1)i}} ) ,

\frac{δ}{δy^{(k)i}} = \frac{∂}{∂y^{(k)i}} .

The set of functions {N_{(1)}^{j}_{i}, ..., N_{(k)}^{j}_{i}} are called the coefficients of the nonlinear

connection N.

The dual adapted basis is as follows

(2.6)  {δx^{i}, δy^{(1)i}, ..., δy^{(k)i}} ,

where

(2.6)′  δx^{i} = dx^{i}, δy^{(1)i} = dy^{(1)i} + M_{(1)}^{i}_{j}dx^{j}, ..., δy^{(k)i} =

= dy^{(k)i} + M_{(1)}^{i}_{j}dy^{(k−1)i} + ... + M_{(k)}^{i}_{j}dx^{j}.

The set of functions {M_{(1)}^{i}_{j}, ..., M_{(k)}^{i}_{j}} are the dual coefficients of the nonlinear

connection N.

The relationships between the direct coefficients (N_{(1)}^{i}_{j}, ..., N_{(k)}^{i}_{j}) and the dual

coefficients of the nonlinear connection N are the following:

(2.7)  \{ \begin{matrix}M_{(1)}^{i}_{j} & = N_{(1)}^{i}_{j}, \\ M_{(2)}^{i}_{j} & = N_{(2)}^{i}_{j} − N_{(1)}^{i}_{m}M_{(1)}^{m}_{j} \\ ......... & ................................................... \\ M_{(k)}^{i}_{j} & = N_{(k)}^{i}_{j} − N_{(k−1)}^{i}_{m}M_{(1)}^{m}_{j} − ... − N_{(1)}^{i}_{m} M_{(k−1)}^{m}_{j}. \end{matrix}

Here we get an important results: Theorem 2.1. (Miron-Atanasiu). A k-spray S with the coefficients G^{i} determines a nonlinear connection N, whose dual coefficients are

M_{(1)}^{i}_{j} = \frac{∂G^{i}}{∂y^{(k)j}} , M_{(2)}^{i}_{j} = \frac{1}{2} (SM_{(1)}^{i}_{j} + M_{(1)}^{i}_{m}M_{(1)}^{m}_{j}) , ...,

(2.8)  M_{(k)}^{i}_{j} = \frac{1}{k} (S M_{(k−1)}^{i}_{j} + M_{(1)}^{i}_{m} M_{(k−1)}^{m}_{j}) .

3. As a first application we solve the problem of prolongation of order k of a Riemann space R^{n} = (M, g).

The problem is to determine a Riemannian metric G on E = Osc^{k}M depending only on the given metric g on the base manifold M.

We shall give here, without demonstrations, the solution of this old and difficult problem.

Let g_{ij}(x) be the local components of the given Riemannian metric g of the space R^{n} and γ_{jh}^{i}(x) be its Christoffel symbols. Theorem 3.1. There exist the nonlinear connections N on E determined only by the given Riemannian structure g. One of them has the dual coefficients

(3.1)  \{ \begin{matrix}M_{(1)}^{i}_{j} & = γ_{jm}^{i}(x)y^{(1)}_{m}, \\ M_{(2)}^{i}_{j} & = \frac{1}{2} (ΓM_{(1)}^{i}_{j} + M_{(1)}^{i}_{m}M_{(1)}^{m}_{j}) , \\ ......... & ............................................. \\ M_{(k)}^{i}_{j} & = \frac{1}{k} (Γ M_{(k−1)}^{i}_{j} + M_{(1)}^{i}_{m} M_{(k−1)}^{m}_{j}) , \end{matrix}

where Γ is the operator

(3.2)  Γ = y^{(1)i} \frac{∂}{∂x^{i}} + ... + ky^{(k)i} \frac{∂}{∂y^{(k−1)}} .

This nonlinear connection N, depending only on g, will be called canonical. Let us consider the dual basis (2.6) adapted to the canonical nonlinear con-

nection N and the tensor field G on E:

(3.3) G = g_{ij}(x)dx^{i} ⊗ dx^{j} + g_{ij}(x)δy^{(1)i} ⊗ δy^{(1)j} + ... + g_{ij}(x)δy^{(k)i} ⊗ δy^{(k)j}.

We can formulate the following important result: Theorem 3.2. The pair P rol^{k}R^{n} = (Osc^{k}M, G) is a Riemann space of dimension (k + 1)n, whose metric G depends only on the given Riemannian metric g of the space R^{n} = (M, g).

With this Theorem the problem of prolongation to Osc^{k}M of the Riemannian space R^{n} is solved.

An important remark: If we represent the Liouville vector fields Γ^{1}, ..., Γ^{k} in the adapted basis (2.5) of the canonical nonlinear connection N, we get, for instance

Γ^{k}= z^{(1)i} \frac{δ}{δy^{(1)i}} + ... + kz^{(k)i} \frac{δ}{δy^{(k)i}} .

In this case z^{(1)i}, ..., z^{(k)i} are d-vector fields. They have the expressions

z^{(1)i} = y^{(1)i}, 2z^{(2)i} = 2y^{(2)i} + M_{(1)}^{i}_{m}y^{(1)m},

(3.4)  kz^{(k)i} = ky^{(k)i} + (k − 1)M_{(1)}^{i}_{m}y^{(k−1)m} + ... + M_{(k−1)}^{i}_{m}y^{(1)m}.

Therefore, the following functions

(3.5)  L(x, y^{(1)}, ..., y^{(k)}) = g_{ij}(x)z^{(k)i}z^{(k)j}

is an important Lagrangian of order k, which depends on the Riemannian structure g, only.

Therefore, the geometry of the space P rol^{k}R^{k} can be identified to the geometry of the higher order Lagrange spaces L^{(k)n} = (M, L), in which the fundamental function is given by (3.5).

Of course, we can formulate the same problem for the prolongation of order k of the Finsler spaces F^{n} or for the Lagrange space L^{n}.

4. A second application can be made in the problem of the geometrization of the variational calculus concerning the functionals of the form:

(4.1)  I(c) = ∫_{0}^{1} L (x, \frac{dx}{dt}, ..., \frac{1}{k!} \frac{d^{k}x}{dt^{k}}) dt,

where L : Osc^{k}M → R is a differentiable Lagrangian of order k.

Remarking that

(4.2)  E^{0}_{i} (L) =^{def} \frac{∂L}{∂x^{i}} − \frac{d}{dt} \frac{∂L}{∂y^{(1)i}} + ... + (−1)^{k} \frac{1}{k!} \frac{d^{k}}{dt^{k}} \frac{∂L}{∂y^{(k)i}}

is a d-covector field along the curve c : [0, 1] → M we prove that the Euler-Lagrange equations derived from (4.1) are given by

(4.2)′  E^{0}_{i} (L) = 0, y^{(1)i} = \frac{dx^{i}}{dt} , ..., y^{(k)i} = \frac{1}{k!} \frac{d^{k}x^{i}}{dt^{k}} .

Now, we can prove: Lemma 4.1. For any differentiable Lagrangian L(x, y^{(1)}, ..., y^{(k)}) and for any function ϕ(t), along a smooth curve c we have

(4.3)  E^{0}_{i} (ϕL) = ϕ E^{0}_{i} (L) + \frac{dϕ}{dt} E^{1}_{i} (L) + ... + \frac{d^{k}ϕ}{dt^{k}} E^{k}_{i} (L),

where E^{0}_{i} (L) is the d-covector field (4.2) and E^{1}_{i} (L), ..., E^{k}_{i} (L) are d-covector fields which depend on the Lagrangian L only.

The d-covectors E^{1}_{i} (L), ..., E^{k}_{i} (L) are called the Craig-Synge covectors.

An important result is expressed by: Theorem 4.1. If the Lagrangian L(x, y^{(1)}, ..., y^{(k)}) is regular, then the equations

E^{k−1}_{i} (L) = 0 determines a k-spray whose coefficients depend on the Lagrangian L only.

Now let us consider the Lie derivation operators with respect to the Liouville vector fields

(4.4)  I^{1} = £_{Γ^{1}}, ..., I^{k} = £_{Γ^{k}}.

We call energy of order k of the Lagrangian L(x, y^{(1)}, ..., y^{(k)}) with respect to a smooth curve c the scalar field

(4.5) E_{c}^{k}(L) = I^{k}(L) − \frac{1}{2!} \frac{dI^{(k−1)}(L)}{dt} + ... + (−1)^{(k−1)} \frac{1}{k!} \frac{d^{(k−1)}I^{1}(L)}{dt^{(k−1)}} − L.

M. de Leon, D.Krupka et alt, [2], prove the following theorems: Theorem 4.2. For any differentiable Lagrangian L(x, y^{(1)}, ..., y^{(k)}) along a smooth curve c : [0, 1] → (x^{i}(t)) ∈ M we have

(4.6)  \frac{dE_{c}^{k}(L)}{dt} = − E^{0}_{i} (L)\frac{dx^{i}}{dt} .

Theorem 4.3. For any differentiable Lagrangian L(x, y^{(1)}, ..., y^{(k)}) the energy of order k, E_{c}^{k}(L), is conserved along of every solution curve c of the Euler-Lagrange

equation E^{0}_{i} (L) = 0.

There are the energies of order k−1, ..., 1. They are used in a Nöther theorem concerning the higher order Lagrangians, [2].

5. In the category of the geometries of superior order there are these called by us the geometries of the higher order Lagrange spaces, [2].

Let us consider the manifold Ẽ = Õsc^{k}M formed by all points (x, y^{(1)}, ..., y^{(k)}) ∈ E for which y^{(1)} ≠ 0.

Thus, we can define a Lagrange space of order k, as a pair L^{(k)n} = (M, L), where M is a real n-dimensional C^{∞}-manifold and L(x, y^{(1)}, ..., y^{(k)}) is a regular Lagrangian on Ẽ for which the d-tensor field

(5.1)  g_{ij} = \frac{1}{2} \frac{∂^{2}L}{∂y^{(k)i}∂y^{(k)j}}

has a constant signature.

L is called the fundamental function of the space L^{(k)n} and g_{ij} its fundamental tensor field.

The condition of regularity is expressed by

(5.1)′  rank ∥ g_{ij} ∥= n onẼ.

An example of Lagrange space of order k is given by the space L^{(k)n} endowed with the fundamental function L from (3.5)

Its fundamental tensor field is just the metric tensor g_{ij}(x) of the Riemannian space R^{n} = (M, g).

More general, the Lagrangian

L(x, y^{(1)}, ..., y^{(k)}) = g_{ij}(x)z^{(k)i}z^{(k)j}+

(5.2)  +a_{i}(x, y^{(1)}, ..., y^{(k−1)})z^{(k)i} + b(x, y^{(1)}, ..., y^{(k−1)}),

where the first term is from (3.5) and a_{i}(x, y^{(1)}, ..., y^{(k−1)}) is a covector field with the property \frac{∂a_{i}}{∂y^{(k)j}} = 0 and b(x, y^{(1)}, ..., y^{(k−1)}) is a scalar field with the same property \frac{∂b}{∂y^{(k)i}} = 0, determines a space L^{(k)n}.

The Lagrangian (5.2) is a direct extenssion to Osc^{k}M of that from the classical electrodynamics, [1,2,3].

The geometry of a space L^{(k)n} = (M, L) is based on the canonical k-spray S, determined by the fundamental function L. It is given by theorem 4.1.

Namely, we have: Theorem 5.1. In a Lagrange space of order k, L^{(k)n} = (M, L) there exist the k-sprays which depend only by the fundamental function L. One of them has the coefficients:

(5.3)  (k + 1)G^{i} = \frac{1}{2}g^{ij} \{Γ (\frac{∂L}{∂y^{(k)j}} ) − \frac{∂L}{∂y^{(k−1)j}} \}

where Γ is the operator (3.2).

The k-spray S with the coefficients in (5.3) is called canonical

Using this k-spray, by means of the theorem 2.1, we determine the canonical nonlinear connection N of the space L^{(k)n}.

Therefore, we can prove: Theorem 5.2. 1). There exist N-linear connections D on Osc^{k}M, which depend by the fundamental function L only and have the properties

g_{ij|m} = 0, g_{ij} |^{(α)}_{m}= 0, (α = 1, ..., k).

2). There exists only one metrical N-connection D in L^{(k)n} for which h− and

v^{α} - torsions vanish. Its coefficients CΓ(N) = (L_{jk}^{i}, C_{(α)}_{jk}^{i}) are given by

(5.4)  L_{ij}^{m} = \frac{1}{2}g^{ms} (\frac{δg_{is}}{δx^{j}} + \frac{δg_{js}}{δx^{l}} − \frac{δg_{ij}}{δx^{s}} ) ,

C_{(α)}_{ij}^{m} = \frac{1}{2}g^{ms} (\frac{δg_{is}}{δy^{(α)j}} + \frac{δg_{js}}{δy^{(α)i}} − \frac{δg_{ij}}{δy^{(α)s}} ) , (α = 1, ..., k).

So, CΓ(N), with the coefficients (5.4) is the canonical metrical N-linear connection of the Lagrange space of order k, L^{(k)n}.

The main geometrical properties of the space L^{(k)n} can be derived from the canonical k-spray, canonical nonlinear connection N and from the canonical metrical N-linear connection, CΓ(N).

Finally, we remark that a good geometrical model for the higher order Lagrange space L^{(k)n} is a (k − 1)n-almost contact Riemannian space H^{(k−1)n} = (Osc^{k}M, G, F, α, a, η(α)a) intrinsicly associated to L^{(k)n}, [2]. This space is used in the study of the problem of generalization of physical fields which depend on the higher order velocities.

References

[1] P.L. Antonelli and R. Miron, Lagrange and Finsler geometry. Applications to Physics and Biology, Kluwer Acad.Publ., FTPH no.76, 1995.

[2] R. Miron, The Geometry of Higher-order Lagrange spaces. Applications in Mechanics and Physics (to appear)

[3] R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces. Theory and Applications, Kluwer Acad.Publ., FTPH no.59, 1994.

University Al.I.Cuza Faculty of Mathematics 6600 Iaşi, ROMANIA