DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 321–328

SYMMETRIES OF LAGRANGIAN FIELDS ARE HOMOTHETIES

IN THE HOMOGENEOUS CASE

János Szenthe

DEFINITIONS. Let M be a smooth manifold, O(M) be the zero section of the tangent bundle T M and T̆ M = T M − O(M). Let X : T̆ M → T T M be a 2nd-order differential equation over M, a diffeomorphism φ : M → M is said to be a symmetry of the 2nd-order differential equation X if

(T φ)_{∗}X = T T φ ∘ X ∘ T φ^{−1} = X

holds. Consider the canonical projection π_{M} : T M → M and an integral curve ξ : I → T M of X, then π_{M} ∘ ξ : I → M is called a base integral curve of X. A diffeomorphism φ : M → M is a symmetry of X if and only if it maps the set of base integral curves of X onto itself. In the particular case when X is a spray, in other words, when L_{A}X = X holds, where A : T M → T T M is the Liouville field on T M, then φ is called a symmetry of the spray. Subsequently, the case is considered when X is obtained as a Lagrangian field of a Lagrangian function L : T M → R, which means that X satisfies the Euler-Lagrange Equation

ι_{X}dd_{υ} = d(L − AL)

where d_{υ} is the vertical differential (see e.g. [Be] pp 14-28 and [Go] pp 169-177). In such a case (M, L, X) is called a Lagrangian system which is said to be regular if the 2-form dd_{υ}L is regular. If L is homogeneous of degree k, in other words, if AL = kL holds then (M, L, X) is said to be homogeneous of degree k. If (M, L, X) is a regular homogeneous Lagrangian system then X is a spray (see e.g. [Sz1]).

In particular, let (M, g) be a semi-Riemannian manifold and X : T M → T T M the spray associated with its Levi-Cività covariant derivation ∇. Then X is also a Lagrangian field associated with the Lagrangian function L : v ↦ g(v, v), v ∈ T M (see e.g. [Be] pp 28-32). Moreover, the symmetries of X are affine transformations of (M, g). In fact, if ψ : M → M is an affine transformation of (M, g) then ψ preserves the set of geodesics of ∇ which are the base integral curves of X and consequently ψ is a symmetry of X. Conversely, if a diffeomorphism φ : M → M is a symmetry of X, then φ is an affine transformation of (M, g); in fact, a covariant derivation ∇′ is defined over M by

1991 Mathematics Subject Classification. Primary 58F05; secondary 70H35, 53C60.

Key words and phrases. Sprays, Lagrangian systems, symmetries of Lagrangian systems.

Supported by Hungarian Nat. Found. for Sci. Research Grant No. T17314(1995).

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

∇′_{U}V = T φ^{−1}∇_{T φU}T φV, U, V ∈ T (T M),

which has the same geodesics as ∇ and has also zero torsion; but then ∇ = ∇′ holds; which means that ψ is an affine transformation of (M, g). It is a basic result in Riemannian geometry, that any affine transformation of an irreducible Riemannian manifold is a homothety. An extension of the above result to homogeneous Lagrangian systems is presented below.

DEFINITION. Let (M, L, X) be a Lagrangian system over a smooth manifold M. A diffeomorphism φ : M → M is said to be a homothety of the Lagranian system if

(T φ)^{∗}L = L ∘ T φ = λ · L

holds with some λ ∈ R. In the particular case when L is the energy function of a semi-Riemannian manifold (M, g), in other words, when L(v) = g(v, v), v ∈ T M holds, and a diffeomorphism ψ : M → M is a usual homothety of the semi-Riemannian manifold, in other words, if

g(T ψU, T ψV) = μg(U, V)

holds with some μ ∈ R for U, V ∈ T (M), then also ((T ψ)^{∗})L(W) = L(T ψ(W)) = g(T ψ(W), T ψ(W)) = μg(W, W) = μL(W) holds for W ∈ T (M), which shows that ψ is a homothety of the corresponding Lagrangian system (M, L, X) as well. Assume conversely that φ : M → M is a homothety of the Lagrangian system (M, L, X) then g(T φ(W), T φ(W)) = ((T ϕ)^{∗}L)(W) = λ · L(W) = λ · g(W, W) holds for any W ∈ T (M); but then g(T φU, T φV) = λ · g(U, V) is obviously valid for any U, V ∈ T (M) which implies that φ is a usual homothety of the semi-Riemannian manifold (M, g) too. A homothety of a regular Lagrangian system obviously yields a symmetry of its Lagrangian field as well.

It is a fundamental fact that a 2nd-order differential equation X : T̆ M → T T M defines a connection by the definition γ = −L_{X}υ where υ is the vertical endomorphism (see e.g. [K]). The connection form γ defines a derivation ι_{γ} of degree 0 of the algebra ∧(T̆ M) by the definition:

ι_{γ}ξ : (U_{1}, ..., U_{k}) ↦ ∑_{i} ξ(U_{1}, ..., U_{i−1}, γU_{i}, U_{i+1}, ..., U_{k})

where ξ ∈ ∧^{k}(T̆ M). But then ι_{γ} yields a derivation d_{γ} of degree 1 by the definition d_{γ} = [ι_{γ}, d] as obvious calculations show.

LEMMA 1. Let X : T̆ M → T T M be a 2nd-order differential equation over a smooth manifold M and γ = −L_{X}υ the connection form defined by X. Then

d_{γ} = [d_{υ}, L_{X}]

holds where d_{υ} is the vertical differential over the smooth manifold M.

Proof. It is sufficient to show that the differential operators d_{γ} and [d_{υ}, L_{X}] have the same effect on any 0-form ξ = F and 1-form ξ = dF where F ∈ F(M) (see e.g. [Go] pp 87-89).

1st case: ξ = F. Consider a smooth vector field V ∈ T (T̆ M), then the following hold

(d_{γ})(v) = dF (γV) = dF (−L_{X}υV) =
= −dF ([X, υV] − υ[X, V]) = −X((υV)F) + (υV)(XF) + υ[X, V]F.

On the other hand

([d_{υ}, L_{X}]F)V = (d_{υ}(XF) − L_{X}ι_{υ}dF)V =
= (υV)(XF) − ι_{V} L_{X}ι_{υ}dF =
= (υV)(XF) − X((υV)F) + (ι_{υ}dF)([X, V]).

Thus the identity is valid in the case considered.

2nd case: ξ = dF. The following are valid now:

d_{γ}dF = (ι_{γ}d − dι_{γ})dF = −dι_{γ}dF =
= −d(ι_{γ}d − dι_{γ})F = −dd_{γ}F,
[d_{υ}, L_{X}]dF = (d_{υ}L_{X} − L_{X}d_{υ})dF =
= d_{υ}dL_{X} + L_{X}dd_{υ}F == −d(d_{υ}L_{X} − L_{X}d_{υ})F =
= −d[d_{υ}, L_{X}]F = [d_{υ}, L_{X}]dF.

Thus the validity of the identity in the second case follows now by that of the first one. ♠

The connection γ yields the horizontal and vertical projection fields χ, ϖ : T T̆ M → T T̆ M such that γ = χ − ϖ holds (see e.g. [K]).

LEMMA 2. Let (M, L, X) be a regular Lagrangian system over a smooth manifold M such that L is a first integral of X. If χ : T T M → T T M is the horizontal projection field associated with the connection form γ = −L_{X}υ, then d_{χ}L = 0 holds for the antiderivation d_{χ} = [ι_{χ}, d] = ι_{χ}d − dι_{χ} associated with χ.

Proof. Let F : T̆ M → R be an arbitrary smooth function. Then considering the above expression of the horizontal projection field, the preceding Lemma and the Cartan Identity, the following is obtained

2d_{χ}F = (d + d_{γ})F = dF + [d_{υ}, L_{X}]F =

= dF + d_{υ}(dι_{X} + ι_{X}d)F − (dι_{X} + ι_{X}d)d_{υ}F =
= dF + d_{υ}ι_{X}dF − d(AF) − ι_{X}dd_{υ}F =
= d(F − AF) − ι_{X}dd_{υ}F + d_{υ}(XF),

where A is the Liouville field and the equality υX = A was also applied. Therefore, in the particular case when F = L the above identity and the Euler-Lagrange Equation yield that

2d_{χ}L = d_{υ}(XL)

holds. Since L is a first integral of X, the assertion of the Lemma directly follows. ♠

COROLLARY. (J. Klein) Let (M, L, X) be a regular Lagrangian system which is homogeneous of degree k where k ≠ 1 then d_{χ}L = 0 holds where χ is the horizontal projection field associated with the connection γ = −L_{X}υ.

Proof. In fact, the Euler Lagrange Equation has now the following form

ι_{X}dd_{υ}L = (1 − k)dL

and therefore (1 − k)ι_{X}dL = (1 − k)XL = 0 holds and thus XL = 0 is obtained. But then the preceding Lemma applies. ♠

LEMMA 3. Let (M, L, X) be a regular Lagrangian system such that L is homogeneous of degree k where k ≠ 1. Then (M, L^{p}, X) is a regular Lagrangian system as well for any p > 0.

Proof. In fact, the following equalities are obviously valid:

dd_{υ}(L^{p}) = p((p − 1)L^{p−2}dL ∧ d_{υ}L + L^{p−1}dd_{υ}L),

ι_{X}dd_{υ}(L^{p}) = p(p − 1)L^{p−2}(ι_{X}dL · d_{υ}L − ι_{X}d_{υ}L · dL) + pL^{p−1}ι_{X}dd_{υ}L =

= −p(p − 1)kL^{p−1}dL + pL^{p−1}ι_{X}dd_{υ}L,
d(L^{p} − A(L^{p})) = pL^{p−1}dL − kp^{2}L^{p−1}dL =
= −p(p − 1)kL^{p−1}dL + p(1 − k)L^{p−1}dL.

But then the assertion of the Lemma directly follows. ♠

The assertion of the preceding Lemma also follows from Proposition 1 of [Sz1] where the assumption that L, L′ are first integrals of X was omitted.

DEFINITIONS. Let X : T M → T T M be a spray over a smooth manifold M then the corresponding connection form γ = −L_{X}υ is homogeneous (see e.g. [K]). Then the curvature field R of the connection γ is defined by

R(U, V) = −ϖ[χU, χV], U, V ∈ T (T̆ M).

If v ∈ T_{z}M is a fixed tangent vector then the subspace R_{v} ⊂ V(T M) spanned by the vectors R(U, V)_{v}, U, V ∈ T (T̆ M) is called the image of the curvature tensor at v. If the dimension of the subspaces R_{v}, v ∈ T̆_{z}M is a constant function on T̆_{z}M then its value r_{z} is called the rank of X at z ∈ M (see [K]).

PROPOSITION 4. (J. Klein) If the spray X is Lagrangian field of a Lagrangian function L : T M → R which is homogeneous of degree k ≠ 1 and such that its rank r_{z} exists for z ∈ M, then r_{z} ≤ m − 1 holds where m = dimM.

Proof. In fact, it is sufficient to show that dL(R(U, V)) = 0 holds for any U, V ∈ T (T̆ M). Actually, since d_{χ}L = 0 holds by the Corollary to Lemma 2, the following are valid too

dL(R(U, V))) = dL(ϖ[χU, χV]) =
= dL([χU, χV] − χ[χU, χV]) = dL([χU, χV]) =
= (χU)((χV)L) − (χV)((χU)L) = 0.

But then the assertion follows by the observation above. ♠

DEFINITIONS. Let X : T M → T T M be a spray over a smooth manifold M and γ = −L_{X}υ the corresponding connection form which is homogeneous. The connection form γ defines the horizontal distribution

H = {H_{v}|v ∈ T M}

where H_{v} = {w ∈ T_{v}T M|ϖ(w) = 0} and ϖ is the vertical projection corresponding to the connection form γ (see e. g. [K]). The horizontal distribution H in turn defines for each piecewise smooth loop λ based at z ∈ M the parallel translation π_{λ} : T_{z}M → T_{z}M along λ which proves to be a homogeneous diffeomorhism of T̆_{z}M (see [Sz2]). The above parallel translations form the holonomy group Φ_{z} defined by the distribution at z.

PROPOSITION 5. Let (M, L, X) be a homogeneous and regular Lagrangian system such that the rank of X exits and has maximal value everywhere on M. Then the orbits of the holonomy group Φ_{z} of X are codimension 1 submanifolds of T̆_{z}M for any z ∈ M.

The proof of the above Proposition is given in [Sz2].

THEOREM 6. Let M be smooth m-dimensional manifold and (M, L, X) a regular Lagrangian system where L is homogeneous of degree k with k ≠ 0, 1 and the rank of X exits and has maximal value everywhere on M. Then any diffeomorphism φ : M → M which yields a symmetry of X is a homothety of L as well.

Proof. Observe first that there is no loss of generality by assuming that L is everywhere non-negative on T M. In fact, assume that the Theorem is true for sytems with everywhere non-negative Lagrangians and consider an arbitrary Lagrangian system (M, L, X) which satisfies the conditions of the Theorem. Then by Lemma 3 a Lagrangian system (M, L^{2}, X) is obtained which also satisfies the conditions of the Theorem and which is homogeneous of degree 2k. If φ : M → M yields a symmetry of X, then φ is a homothety of L^{2} by the assumption. In other words

((T φ)^{∗}L(v))^{2} = ((T φ)^{∗}L^{2})(v) = λ · L^{2}(v) = (\sqrt{λ}L(v))^{2}

holds for v ∈ T M. But then φ is obviously a homothety of L as well.

Observe secondly that all the values of L on T̆M are regular. In fact the Euler-Lagrange Equation

ι_{X}dd_{υ}L = (1 − k)dL

and the regularity of L yield that X has to vanish at a sigular point of L. But X being a 2nd-order differential equation cannot vanish on T̆M.

Consider now a Lagrangian system which satisfies the conditions of the Theorem and where the Lagrangian L is everywhere non-negative. Then take first a smooth loop ξ : [0, 1] → M with ξ(0) = ξ(1) = z ∈ M, a tangent vector v ∈ T̆_{z}M and the vector v′ = π_{ξ}v ∈ T_{z}M obtained by the parallel translation π_{ξ} of v along ξ. It will be shown first that

L(v) = L(v′)

is valid. In fact, consider the horizontal projection field χ associated with the connection γ and let ξ̂: [0, 1] → T M be the corresponding horizontal lift of the loop ξ with ξ̂(0) = v. Then ξ̂(1) = v′ holds; moreover,

d(L ∘ ξ̂) = dL(˙ξ̂) = dL(χ˙ξ̂) = d_{χ}L(˙ξ̂) = 0

holds, since d_{χ}L = 0 is valid according to the Corollary to Lemma 2. Consequently, the following holds as well:

L(v′) = L ∘ ξ̂(1) = L ∘ ξ̂(0) = L(v).

Assume now that the smooth loop ξ is embedded into a smooth family ξ_{θ}, θ ∈ [0, 1] of smooth loops; more precisely, consider a smooth map

Ξ : [0, 1] × [0, 1] → M

such that for each fixed θ ∈ [0, 1] a smooth loop ξ_{θ}(τ) = Ξ(τ, θ), τ ∈ [0, 1] with ξ_{θ}(0) = ξ_{θ}(1) = z is obtained, moreover ξ_{0}(τ) = z for τ ∈ [0, 1] and ξ_{1} = ξ. Let now v′_{θ} = π_{ξ_{θ}}v be the vector obtained by parallel translation of v along ξ_{θ} where θ ∈ [0, 1]. Then a smooth curve

θ ↦ v′_{θ} ∈ T_{z}M, θ ∈ [0, 1]

is obtained, where v′_{0} = v and v′_{1} = v′. Moreover, the above curve lies in that level set L_{v} of the function L which contains v by the preceding observation. Consequently, the tangent vector

\frac{d}{dθ} v′_{θ}|_{θ=0} ∈ T_{v}T_{z}M

of the above curve is in the tangent space T_{v}L_{v} of the level set L_{v} which is a submanifold. The preceding observations obviously extend to the case of piecewise smooth loops. On the other hand, the orbit Φ_{z}(v) of the holonomy group is a codimension 1 submanifold of T̆_{z}M by Proposition 5, and therefore any tangent vector of the orbit Φ_{z}(v) at v is obtainable as a linear combination of tangent vectors of curves of the above kind. Therefore, the inclusion

T_{v}Φ_{z}(v) ⊂ T_{v}L_{v}

holds. Furthermore, for any horizontal smooth curve η : I → T M the equality η̇L = d_{χ}L(η̇) = 0 is valid by Lemma and consequently even

H_{v} ⊂ T_{v}L_{v}

holds since the level set L_{v} is a smooth codimension 1 submanifold for v ∈ T̆ M. But then

2m − 1 = dimT_{v}L_{v} ≥ dim(H_{v} ⊕ T_{v}Φ_{z}(v)) ≥ 2m − 1

holds since T_{v}Φ_{z}(v) is included in the vertical subspace V_{v} and since dimT_{v}Φ_{z}(v) = m − 1 by assumptions of the theorem. But then

T_{v}L_{v} = H_{v} ⊕ T_{v}Φ_{z}(v)

is valid in consequence of the preceding inequalities.

By basic observations (M, (T φ)^{∗}L, (T φ)_{∗}X) is a regular Lagrangian system which is homogeneous of degree k as well. Moreover, (T φ)_{∗}X = X holds by assumption. Consequently, the preceding observations applied to the Lagrangian system (M, (T φ)^{∗}L, X) yield that

T_{v}(T φ^{∗}L)_{v} = H_{v} ⊕ T_{v}Φ_{z}(v)

holds, provided that v ⊂ T̆ M is valid. Therefore, the equality

T_{v}L_{v} = T_{v}((T φ)^{∗}L)_{v}

holds. But then the regular level sets of L and (T φ)^{∗}L coincide.

Consider now the restricted functions L⌈T_{z}M and (T φ)^{∗}L⌈T_{z}M in case of a z ∈ M. The above functions cannot have the value 0 on T̆M; in fact, since the orbits of the holonomy group are included in level sets of the above functions and these orbits intersect each 1-dimensional subspace of T_{z}M, the homogeneity of the above functions implies that the value 0 can be only singular; but the above functions cannot have singular value on T̆ M by a former observation. Since both the functions L⌈T_{z}M and (T φ)^{∗}L⌈T_{z}M are homogeneous of degree k ≠ 0 and everywhere non-negative, a ray R^{+}v, v ∈ T̆_{z}M intersects each level set of L and of (T φ)^{∗}L in a strictly monotone way. But then there is a function λ_{z} : R^{+} → R^{+} such that

((T φ)^{∗}L)⌈T_{z}M = λ_{z} ∘ (L⌈T_{z}M)

holds and the function λ_{z} is smooth. But then by homogeneity of the restricted functions

ξ^{k}λ_{z}(L(v)) = ξ^{k}((T φ)^{∗}L(v)) = ((T φ)^{∗}(ξ^{k}L(v)) = (T φ)^{∗}L)(ξv) =
= λ_{z}(L(ξv)) = λ_{z}(ξ^{k}L(v))

holds for any ξ ∈ R which implies that the function λ_{z} is homogeneous of degree 1. In other words,

λ_{z}(τ) = σ(z) · τ, τ ∈ R

holds with some σ(z) ∈ R and consequently, ((T φ)^{∗}L)(v) = L(σ(z)v), v ∈ T_{z}M is valid too.

Considering the definition of a homothety of L, it is sufficient now to show that the function z ↦ σ(z), z ∈ M is constant. In fact, consider the pull back σ̃ : T M → R of the above function by π_{M}. Since L and (T φ)^{∗}L have the same level sets the function σ̃ is constant on each of these common level sets. But then σ̃ itself must be constant. Consequently, σ is constant as well. ♠

References

[Be] Besse, A., Manifolds all of whose Geodesics are Closed. Berlin-Heidelberg-New York, 1978

[Go] Godbillon, C., Géométrie Différentielle et Mécanique Analytique. Paris, 1969 [Kl] Klein, J., Geometry of sprays, Proc. of the IUTAM-ISIMM Symposium of

Torino, 177-193

[Sz1] Szenthe, J., Lagrangians and sprays, Ann. Univ. Sci. Budapest 35(1992), 103-107

[Sz2] Szenthe, J., On the holonomy groups of homogeneous connections. (To appear)

Department of Geometry Eötvös University, Budapest Rákóczi út 5, H-1088, HUNGARY