Projective Metrizability and Formal Integrability
The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order part...
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irk-123456789-1480912019-02-17T01:25:46Z Projective Metrizability and Formal Integrability Bucataru, I. Muzsnay, Z. The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential operator P₁ and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P₁ using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P₁ is involutive and hence one of the two conditions is always satisfied. While discussing the second condition, in Theorem 4.3 we prove that there is only one obstruction to the formal integrability of P₁, and this obstruction is due to the curvature tensor of the induced nonlinear connection. When the curvature obstruction is satisfied, the projective metrizability problem reduces to the discussion of the algebraic conditions, which as we show are always satisfied in the analytic case. Based on these results, we recover all classes of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are projectively metrizable without being Finsler metrizable. 2011 Article Projective Metrizability and Formal Integrability / I. Bucataru, Z. Muzsnay // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 49N45; 58E30; 53C60; 58B20; 53C22 DOI: http://dx.doi.org/10.3842/SIGMA.2011.114 http://dspace.nbuv.gov.ua/handle/123456789/148091 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential operator P₁ and a set of algebraic conditions on semi-basic 1-forms. We discuss the formal integrability of P₁ using two sufficient conditions provided by Cartan-Kähler theorem. We prove in Theorem 4.2 that the symbol of P₁ is involutive and hence one of the two conditions is always satisfied. While discussing the second condition, in Theorem 4.3 we prove that there is only one obstruction to the formal integrability of P₁, and this obstruction is due to the curvature tensor of the induced nonlinear connection. When the curvature obstruction is satisfied, the projective metrizability problem reduces to the discussion of the algebraic conditions, which as we show are always satisfied in the analytic case. Based on these results, we recover all classes of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are projectively metrizable without being Finsler metrizable. |
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Bucataru, I. Muzsnay, Z. |
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Bucataru, I. Muzsnay, Z. Projective Metrizability and Formal Integrability Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Bucataru, I. Muzsnay, Z. |
| author_sort |
Bucataru, I. |
| title |
Projective Metrizability and Formal Integrability |
| title_short |
Projective Metrizability and Formal Integrability |
| title_full |
Projective Metrizability and Formal Integrability |
| title_fullStr |
Projective Metrizability and Formal Integrability |
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Projective Metrizability and Formal Integrability |
| title_sort |
projective metrizability and formal integrability |
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Інститут математики НАН України |
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2011 |
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http://dspace.nbuv.gov.ua/handle/123456789/148091 |
| citation_txt |
Projective Metrizability and Formal Integrability / I. Bucataru, Z. Muzsnay // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 32 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
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AT bucatarui projectivemetrizabilityandformalintegrability AT muzsnayz projectivemetrizabilityandformalintegrability |
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2025-07-12T18:13:27Z |
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2025-07-12T18:13:27Z |
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1837465884920119296 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 114, 22 pages
Projective Metrizability and Formal Integrability
Ioan BUCATARU † and Zoltán MUZSNAY ‡
† Faculty of Mathematics, Al.I.Cuza University, B-dul Carol 11, Iasi, 700506, Romania
E-mail: bucataru@uaic.ro
URL: http://www.math.uaic.ro/~bucataru/
‡ Institute of Mathematics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary
E-mail: muzsnay@science.unideb.hu
URL: http://www.math.klte.hu/~muzsnay/
Received August 25, 2011, in final form December 08, 2011; Published online December 12, 2011
http://dx.doi.org/10.3842/SIGMA.2011.114
Abstract. The projective metrizability problem can be formulated as follows: under what
conditions the geodesics of a given spray coincide with the geodesics of some Finsler space,
as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for
a spray in terms of a first-order partial differential operator P1 and a set of algebraic con-
ditions on semi-basic 1-forms. We discuss the formal integrability of P1 using two sufficient
conditions provided by Cartan–Kähler theorem. We prove in Theorem 4.2 that the symbol
of P1 is involutive and hence one of the two conditions is always satisfied. While discussing
the second condition, in Theorem 4.3 we prove that there is only one obstruction to the
formal integrability of P1, and this obstruction is due to the curvature tensor of the in-
duced nonlinear connection. When the curvature obstruction is satisfied, the projective
metrizability problem reduces to the discussion of the algebraic conditions, which as we
show are always satisfied in the analytic case. Based on these results, we recover all classes
of sprays that are known to be projectively metrizable: flat sprays, isotropic sprays, and
arbitrary sprays on 1- and 2-dimensional manifolds. We provide examples of sprays that are
projectively metrizable without being Finsler metrizable.
Key words: sprays; projective metrizability; semi-basic forms; partial differential operators;
formal integrability
2010 Mathematics Subject Classification: 49N45; 58E30; 53C60; 58B20; 53C22
1 Introduction
The projective metrizability problem for a homogeneous system of second-order ordinary diffe-
rential equations, which can be identified with a spray S, seeks for a Finsler metric F whose
geodesics coincide with the geodesics of the spray S, up to an orientation preserving reparame-
terization. For the case when S is a flat spray this problem was first studied by Hamel [16]
and it is known as the Finslerian version of Hilbert’s fourth problem [1, 11, 30]. In the general
case it was Rapcsák [27] who obtained, in local coordinates, necessary and sufficient condi-
tions for the projective metrizability problem of a spray. Global formulations for the projec-
tive metrizability problem where obtained by Klein and Voutier [17], and by Szilasi and Vat-
tamány [31]. It has been shown that this is an essential problem in various fields of biology and
physics [3].
The projective metrizability problem can be formulated as a particular case of the inverse
problem of the calculus of variations. We refer to [2, 8, 19, 25, 28] for various approaches of
the inverse problem of the calculus of variations. One of this approaches seeks for the exis-
tence of a multiplier matrix that satisfies four Helmholtz conditions [19, 28]. In [5], these four
Helmholtz conditions where reformulated in terms of a semi-basic 1-form. For the particular
mailto:bucataru@uaic.ro
http://www.math.uaic.ro/~bucataru/
mailto:muzsnay@science.unideb.hu
http://www.math.klte.hu/~muzsnay/
http://dx.doi.org/10.3842/SIGMA.2011.114
2 I. Bucataru and Z. Muzsnay
case of the projective metrizability problem, it has been shown in [5] that only two of the four
Helmholtz conditions are independent. In this work we discuss the formal integrability of these
two Helmholtz conditions using two sufficient conditions provided by Cartan–Kähler theorem.
The approach in this work follows the one developed in [26] for studying the Finsler metrizability
problem for a spray.
In Section 2 we recall first some basic aspects of the Frölicher–Nijenhuis theory on a mani-
fold M [13, 18]. Then, we use this theory on TM and apply it to the natural objects that live on
the tangent space: vertical distribution, Liouville vector field, and semi-basic forms [14, 15, 20].
In Section 3 we use the geometric setting developed in the previous section to reformulate
the projective metrizability problem. In Theorem 3.8 we obtain a set of necessary and sufficient
conditions, for the projective metrizability problem of a spray, which consists of a set of alge-
braic equations (3.7) and a set of differential equations (3.8) on semi-basic forms. The set of
differential equations determine a first-order partial differential operator P1, called the projec-
tive metrizability operator, which acts on semi-basic 1-forms.
In Section 4 we discuss the formal integrability of the projective metrizability operator P1,
using two sufficient conditions provided by Cartan–Kähler theorem. Based on this theorem and
Theorems 4.2 and 4.3 we conclude that there is only one obstruction to the formal integrability
of P1. This obstruction is expressed in terms of the curvature tensor of the nonlinear connection
induced by the spray. In this work we pay attention to various cases when the obstruction
condition is automatically satisfied. Another possibility, which we leave for further work, is to
add this obstruction to the projective metrizability operator and discuss the formal integrability
of the new operator. Using different techniques, an alternative expression of the obstruction
condition was obtain in [31, Theorem 4.9].
In Section 5 we discuss some classes of sprays for which the curvature obstruction is automat-
ically satisfied: flat sprays, isotropic sprays, and arbitrary sprays on 2-dimensional manifolds.
For each of these classes of sprays, the projective metrizability problem reduces to the discussion
of the algebraic conditions (3.7), which as we show are always satisfied in the analytic case. Al-
though, for these classes, the projective metrizability problem has been discussed before by some
authors, our approach in this work is different. Using different methods, it was demonstrated
in [9] that flat sprays are projectively metrizable. In [10] it has been shown that isotropic sprays
are projectively equivalent with flat sprays, and hence are projectively metrizable. On a 2-
dimensional manifold it has been shown by Matsumoto that every spray is projectively related
to a Finsler space [22], by extending the original discussion of Darboux [12] about second-order
differential equations.
We use a spray on a 2-dimensional, considered by Anderson and Thompson in [2], and
a projectively flat spray of constant flag curvature, considered by Yang in [32], to provide
examples of projectively metrizable sprays that are not Finsler metrizable.
2 Preliminaries
In this section we present the differential geometric tools we need to formulate and study the
projective metrizability problem.
A systems of second-order ordinary differential equations on a manifold M can be iden-
tified with a second-order vector field that is called a semispray. To each semispray one
can associate a geometric apparatus very useful to obtain qualitative information regarding:
the variations of its geodesics, their stability, as well as the inverse problem of the calcu-
lus of variations, [4]. A global formulation for the geometric apparatus one can associate to
a semispray is due to Grifone [14] and can be obtained using the Frölicher–Nijenhuis theo-
ry [13].
Projective Metrizability and Formal Integrability 3
2.1 Frölicher–Nijenhuis theory
In this subsection we recall and extend some aspects of the Frölicher–Nijenhuis theory, which
will be applied in the next subsection to vector valued differential forms on tangent bundles. For
the classic and modern formulations of Frölicher–Nijenhuis theory we refer to [13, 14, 15, 18, 20].
In this work M is a real, n-dimensional, smooth manifold. We denote by C∞(M), the ring
of smooth functions on M , and by X(M), the C∞(M)-module of vector fields on M . Consider
Λ(M) =
⊕
k∈N Λk(M) the graded algebra of differential forms on M . We denote by Sk(M) the
space of symmetric (0, k) tensors on M . We also write Ψ(M) =
⊕
k∈N Ψk(M) for the graded
algebra of vector-valued differential forms on M .
For L ∈ Ψl(M), a vector valued l-form, we consider τL : Λ1(M) ⊗ Λk(M) → Λk+l(M), or
τL : Ψ1(M)⊗Ψk(M)→ Ψk+l(M), the alternating operator defined as
(τLB)(X1, . . . , Xk+l) =
1
k!l!
∑
σ∈Sk+l
ε(σ)B(L(Xσ(1), . . . , Xσ(l)), Xσ(l+1), . . . , Xσ(l+k)), (2.1)
where X1, . . . , Xk+l ∈ X(M) and Sk+l is the permutation group of {1, . . . , k + l}.
The restriction of τL to Λk+1(M) ⊂ Λ1(M) ⊗ Λk(M), or the restriction to Ψk+1(M), is
a derivation of degree (l − 1) and it coincides with the inner product iL, see [15, 18]. Inner
product iL is trivial on Λ0(M) = C∞(M), or Ψ0(M) = X(M), and hence it is a derivation
of type i∗ [15] or an algebraic derivation [18]. Since it satisfies the Leibniz rule, iL is uniquely
determined by its action on Λ1(M), or Ψ1(M), when it is given by iLB = B◦L. For the particular
case when l = 1 and L = Id we have that iIdB = kB for all B ∈ Λk(M), or B ∈ Ψk(M).
For a linear connection ∇ on M consider d∇ : Ψk(M) → Ψk+1(M) the covariant exterior
derivative, see [18, § 11.13], given by
d∇B(X1, . . . , Xk+1) =
k+1∑
i=1
(−1)i+1∇XiB(X1, . . . , X̂i, . . . , Xl+1)
+
∑
1≤i<j≤k+1
(−1)i+jB([Xi, Xj ], X1, . . . , X̂i, . . . , X̂j , . . . , Xk+1). (2.2)
The exterior derivative d : Λk(M) → Λk+1(M) satisfies also formula (2.2) for B ∈ Λk(M).
Therefore, we will use the notation d∇ to refer to both, the covariant exterior derivative, or the
exterior derivative. For the latter case d = d∇ does not depend on the linear connection ∇.
For a vector valued l-form L, consider the commutator of the inner product iL and the
(covariant) exterior derivative d (d∇). This differential operator is denoted by dL : Λk(M) →
Λk+l(M) (d∇L : Ψk(M)→ Ψk+l(M)), it is given by
d∇L = iL ◦ d∇ + (−1)ld∇ ◦ iL, (2.3)
it is a derivation of degree l, which is called the (covariant) exterior derivative with respect to L.
Derivation dL (d∇L ) commutes with the exterior derivative d (d∇) and hence it is a derivation
of type d∗ [15] or a Lie derivation [18]. Since it satisfies the Leibniz rule, dL (d∇L ) is uniquely
determined by its action on Λ0(M) = C∞(M) (Ψ0(M) = X(M)). For the particular case when
l = 1 and L = Id we have that d∇Id = d∇. Therefore, we obtain d∇ Id = T , where T is the torsion
of the linear connection ∇.
For two vector valued forms L ∈ Ψl(M) and K ∈ Ψk(M), their Frölicher–Nijenhuis bracket
[L,K] is a vector valued (k + l)-form, defined by
d[L,K] = dL ◦ dK − (−1)kldK ◦ dL. (2.4)
4 I. Bucataru and Z. Muzsnay
For a vector valued l-form L and a linear connection ∇ on M we obtain a derivation of
degree l given by DL = τL∇. Hence, DL : Λk(M) → Λk+l(M) (DL : Ψk(M) → Ψk+l(M)), acts
on (vector-valued) k-forms as follows:
(DLB) (X1, . . . , Xk+l) =
1
l!k!
∑
σ∈Sk+l
ε(σ)
(
∇L(Xσ(1),...,Xσ(l))B
) (
Xσ(l+1), . . . , Xσ(k+l)
)
. (2.5)
For the particular case when l = 1 and L = Id, we denote the corresponding derivation of
degree 1 by D = DId. Since any derivation of degree l can be uniquely decomposed into a sum
of a Lie derivation and an algebraic derivation [18, § 8.3], we obtain for DL the following result.
Lemma 2.1. For a vector valued l-form L and a linear connection ∇ on M , derivation of
degree l, DL, decomposes uniquely into a sum of a Lie derivation and an algebraic derivation as
follows:
DL = d∇L − id∇L Id. (2.6)
Proof. When acting on forms, formula (2.6) reads DL = dL− id∇L Id. The vector valued (l+ 1)-
form that defines the inner product in formula (2.6) is, according to formula (2.3), given by
d∇L Id = iLT + (−1)ld∇L.
Since Lie derivations commute with the exterior derivative d∇ and satisfy the Leibnitz rule
it follows that they are uniquely determined by their action on Λ0(M) = C∞(M) (Ψ0(M) =
X(M)). Using formulae (2.3) and (2.5) one can immediately check that DLf = dLf for any
scalar (vector valued) 0-form f .
Since algebraic derivations are trivial on Λ0(M) = C∞(M) (Ψ0(M) = X(M)), and satisfy the
Leibnitz rule it follows that they are uniquely determined by their action on Λ1(M) (Ψ1(M)).
To prove formula (2.6) we have to show that(
DL − d∇L
)
ω = −ω ◦
(
d∇L Id
)
,
for any (vector valued) 1-form ω. Since formally, we have the same formulae to define the action
on scalar, or vector valued forms, we will work with scalar forms.
Let ω ∈ Λ1(M) and X1, . . . , Xl+1 ∈ X(M). For k = 1, from formula (2.5) we obtain the
action of the derivation DL on 1-forms as follows:
(DLω) (X1, . . . , Xl+1) =
l+1∑
i=1
(−1)l+1−i
(
∇L(X1,...,X̂i,...,Xl+1)
ω
)
(Xi)
=
l+1∑
i=1
(−1)l+1−i
{
L(X1, . . . , X̂i, . . . , Xl+1)(ω(Xi))− ω
(
∇L(X1,...,X̂i,...,Xl+1)
Xi
)}
.(2.7)
From formula (2.3) we obtain that the action of the exterior derivative dL on a 1-form ω is
given by dLω = iLdω + (−1)ld(ω ◦ L). Therefore, for X1, . . . , Xl+1 ∈ X(M) we have
(dLω)(X1, . . . , Xl+1) =
l+1∑
i=1
(−1)l+1−iL(X1, . . . , X̂i, . . . , Xl+1)(ω(Xi))
+
l+1∑
i=1
(−1)l+1−iω([Xi, L(X1, . . . , X̂i, . . . , Xl+1)]) (2.8)
+
∑
1≤i<j≤l+1
(−1)l+i+jω(L([Xi, Xj ], X1, . . . , X̂i, . . . , X̂j , . . . , Xl+1)).
Now, we evaluate d∇L Id = iLT + (−1)ld∇L on l + 1 vectors X1, . . . , Xl+1 ∈ X(M).
Projective Metrizability and Formal Integrability 5
For k = 1, if we restrict the action of τL given by formula (2.1) to Ψ2(M) we obtain that the
inner product iL : Ψ2(M)→ Ψl+1(M) is given by:
(iLT ) (X1, . . . , Xl+1) =
l+1∑
i=1
(−1)l−iT (Xi, L(X1, . . . , X̂i, . . . , Xl+1)). (2.9)
Using formula (2.2), the action of the exterior covariant derivative d∇ on the vector valued
l-form L is given by
d∇L(X1, . . . , Xl+1) =
l+1∑
i=1
(−1)i+1∇XiL(X1, . . . , X̂i, . . . , Xl+1)
+
∑
1≤i<j≤l+1
(−1)i+jL([Xi, Xj ], X1, . . . , X̂i, . . . , X̂j , . . . , Xl+1). (2.10)
Using formulae (2.7), (2.8), (2.9), and (2.10) it follows that
(DLω − dLω) (X1, . . . , Xl+1) = −
(
ω ◦ (iLT + (−1)ld∇L)
)
(X1, . . . , Xl+1),
for all X1, . . . , Xl+1 ∈ X(M), which means that the decomposition (2.6) is true. �
For the particular case when l = 1 and L = Id, we have that the vector valued 2-form d∇Id Id
reduces to torsion T since iIdT = 2T , d∇ Id = T . Therefore, decomposition (2.6) becomes
D = d− iT .
Remark 2.2. Formula (2.6) shows that the difference of the two derivations dL −DL = id∇L Id
is an algebraic derivation. In other words, if ω ∈ Λk(M) vanishes at some point p ∈M , ωp = 0,
then (DLω)p = (dLω)p. For the particular case when l = 1, this result has been shown in [15,
Proposition 2.5].
2.2 Vertical calculus on TM and semi-basic forms
Consider (TM, π,M), the tangent bundle of the manifold M and (TM \ {0}), π,M) the slashed
tangent bundle, which is the tangent bundle with the zero section removed. The tangent bundle
carries some canonical structures, such as the vertical distribution, the Liouville vector field, and
the vertical endomorphism. The differential calculus associated to these structures, using the
Frölicher–Nijenhuis theory developed in the previous subsection, plays an important role in the
geometry of a system of second-order ordinary differential equations, [4, 5, 14, 15, 17, 20, 24].
The vertical subbundle is defined as V TM = {ξ ∈ TTM, (Dπ)(ξ) = 0}. It induces a vertical
distribution V : u ∈ TM 7→ Vu = V TM ∩ TuTM . This distribution is n-dimensional and it is
integrable, being tangent to the leaves of the natural foliation induced by submersion π. If (xi)
are local coordinates on the base manifold M , we denote by (xi, yi) the induced coordinates
on TM . It follows that yi are coordinates along the leaves of the natural foliation, while xi are
transverse coordinates for the foliation. We denote by Xv(TM) the Lie subalgebra of vertical
vector fields on TM . An important vertical vector field on TM is the Liouville vector field,
which locally is given by C = yi∂/∂yi.
The tangent structure (or vertical endomorphism) is the (1, 1)-type tensor field J on TM ,
which locally can be written as follows:
J =
∂
∂yi
⊗ dxi.
6 I. Bucataru and Z. Muzsnay
Tensor J satisfies J2 = 0 and Ker J = Im J = V TM . Tangent structure J is an integrable
structure since the Frölicher–Nijenhuis bracket vanishes, [J, J ] = 0. As a consequence and using
formula (2.4) we have that d2J = 0.
For the natural foliation induced by submersion π and the corresponding vertical distribution
there are some important classes of forms: basic and semi-basic forms. As we will see in the next
sections, semi-basic forms, vector valued semi-basic forms, and vector valued almost semi-basic
forms are important ingredients to formulate and address the projective metrizability problem.
Definition 2.3. Consider ω ∈ Λ(TM) and L ∈ Ψ(TM).
i) ω is called a basic form if both ω and dω vanish whenever one of the arguments of ω
(respectively dω) is a vertical vector field.
ii) ω is called a semi-basic form if it vanishes whenever one of its arguments is a vertical
vector field.
iii) L is called a vector valued semi-basic form if it takes vertical values and vanishes whenever
one of its arguments is a vertical vector field.
iv) L is called a vector valued almost semi-basic form if it vanishes whenever one of its argu-
ments is a vertical vector field and for every vertical vector field X ∈ Xv(TM) we have
that LXL is a vector valued semi-basic form.
In local coordinates a basic k-form ω on TM can be written as
ω =
1
k!
ωi1...ik(x)dxi1 ∧ · · · ∧ dxik .
For basic forms, the coordinates functions ωi1...ik(x) are basic functions, which means that they
are constant along the leaves of the natural foliation.
Locally, a semi-basic k-form ω on TM can be written as
ω =
1
k!
ωi1...ik(x, y)dxi1 ∧ · · · ∧ dxik . (2.11)
We will denote by Λkv(TM) the set of semi-basic k-forms on TM . A 1-form ω on TM is semi-basic
if and only if iJω = ω ◦ J = 0.
In local coordinates, a vector valued semi-basic l-form L on TM can be written as
L =
1
l!
L
j
i1...il
(x, y)
∂
∂yj
⊗ dxi1 ∧ · · · ∧ dxil . (2.12)
In this work all contravariant or covariant indices, of some tensorial coefficients, that refer to
vertical components will be underlined. We will denote by Ψl
v(TM) the set of vector valued
semi-basic l-forms on TM . A vector valued 1-form L on TM is semi-basic if and only if J ◦L = 0
and iJL = L ◦ J = 0. The tangent structure J is a vector valued semi-basic 1-form.
Locally, a vector valued almost semi-basic l-form L on TM can be expressed as
L =
1
l!
Lji1...il(x)
∂
∂xj
⊗ dxi1 ∧ · · · ∧ dxil +
1
l!
L
j
i1...il
(x, y)
∂
∂yj
⊗ dxi1 ∧ · · · ∧ dxil . (2.13)
For a vector X on TM and a vector valued l-form L on TM , the Frölicher–Nijenhuis bracket
[X,L], defined by formula (2.4), is a vector valued l-form on TM given by:
[X,L](X1, . . . , Xl) = [X,L(X1, . . . , Xl)]−
l∑
i=1
L([X,Xi], X1, . . . , X̂i, . . . , Xl),
Projective Metrizability and Formal Integrability 7
for X1, . . . , Xl vector fields on TM . Using the above formula and the fact that the vertical
distribution is integrable it follows that vector valued semi-basic forms are also almost semi-
basic. This can be seen also from the local expressions (2.12) and (2.13).
Next two lemmas give a good motivation for considering the class of vector valued almost
semi-basic forms. We will also see in Section 4 that the partial differential operator we use to
discuss the projective metrizability problem is defined in terms of some vector valued almost
semi-basic forms.
Lemma 2.4. Let L be a vector valued almost semi-basic l-form on TM . Then, the differential
operator dL preserves semi-basic forms, dL : Λkv(TM)→ Λk+lv (TM).
Proof. Consider a vector valued almost semi-basic l-form L, locally given by formula (2.13),
and a semi-basic k-form ω, locally given by formula (2.11). Since dL is a derivation of degree l,
it follows that the (k + l)-form dLω can be expressed locally as follows
dLω =
1
k!
dL(ωi1···ik) ∧ dxi1 ∧ · · · ∧ dxik
+
1
k!
k∑
j=1
(−1)(j−1)lωi1···ikdx
i1 ∧ · · · ∧ dLdxij ∧ · · · ∧ dxik . (2.14)
Using the assumption that L is a vector valued almost semi-basic form, we show that all terms
in the right hand side of the above formula are semi-basic forms. Since L vanishes whenever
one of its arguments is a vertical vector field it follows that for a function f ∈ C∞(TM),
dLf = iLdf = df ◦ L is a semi-basic l-form. Hence dL(ωi1···ik) are semi-basic l-forms.
We will prove now that dLdx
ij = (−1)lddLx
ij are semi-basic (l + 1)-forms. Using the local
expression (2.13) of L, we have
dLx
ij = iLdx
ij = dxij ◦ L =
1
l!
L
ij
i1...il
(x)dxi1 ∧ · · · ∧ dxil ,
which are basic l-forms. Therefore, ddLx
ij are basic and hence semi-basic (l + 1)-forms. One
can conclude now that all terms in the right hand side of formula (2.14) are semi-basic forms
and hence dLω is a semi-basic (k + l)-form. �
Lemma 2.5. Consider ∇ a linear connection on TM such that ∇J = 0 and a vector valued
almost semi-basic l-form L on TM . Then, the differential operator DL preserves semi-basic
forms, DL : Λkv(TM)→ Λk+lv (TM).
Proof. Using formula (2.6) and Lemma 2.4 we have that the differential operator DL preserves
semi-basic forms if and only if the algebraic derivation of degree l, id∇L Id preserves semi-basic
forms. Hence, we will complete the proof if we show that the vector valued (l+ 1)-form d∇L Id =
iLT + (−1)ld∇L takes vertical values whenever one of its arguments is a vertical vector field.
Here T is the torsion of the linear connection ∇.
Using formulae (2.9) and (2.10) we have
(−1)l(d∇L Id)(X1, . . . , Xl+1) = (−1)l(iLT + (−1)ld∇L )(X1, . . . , Xl+1)
=
l+1∑
i=1
(−1)i+1
{
∇L(X1,...,X̂i,...,Xl+1)
Xi + [Xi, L(X1, . . . , X̂i, . . . , Xl+1)]
}
+
∑
1≤i<j≤l+1
(−1)i+jL([Xi, Xj ], X1, . . . , X̂i, . . . , X̂j , . . . , Xl+1). (2.15)
We will show now that whenever one of the (l+ 1) arguments of d∇L Id is a vertical vector field,
then the right hand side of formula (2.15) is a vertical vector field. Using the fact that d∇L Id
8 I. Bucataru and Z. Muzsnay
is a vector valued (l + 1)-form, we will discuss only the case when X1 is a vertical vector field.
Since L vanishes whenever one of its arguments is a vertical vector field, the nonzero vector field
that remains from the right hand side of formula (2.15), when X1 is a vertical vector field, is
∇L(X2,...,Xl+1)X1 + [X1, L](X2, . . . , Xl+1). (2.16)
The condition ∇J = 0 implies that the linear connection ∇ preserves the vertical distribution
and since X1 is a vertical vector field it follows that ∇L(X2,...,Xl+1)X1 is a vertical vector field
as well. Since L is a vector valued almost semi-basic l-form and X1 is a vertical vector field it
follows that [X1, L] is a vector valued semi-basic form, therefore it takes values into the vertical
distribution and hence [X1, L](X2, . . . , Xl+1) is a vertical vector field. It follows that the vector
field in formula (2.16) is vertical and hence we have completed the proof. �
3 Projective metrizability problem of a spray
A system of homogeneous second-order ordinary differential equations on a manifold M , whose
coefficients do not depend explicitly on time, can be identified with a special vector field on TM
that is called a spray. In this section we address the following question, known as the projective
metrizability problem: for a given spray S find necessary and sufficient conditions for the
existence of a Finsler function F such that the geodesics of S and the geodesics of F coincide
up to an orientation preserving reparameterization. We obtain such necessary and sufficient
conditions in Theorem 3.8 and these conditions are expressed in terms of semi-basic 1-forms.
Particular aspects of the projective metrizability problem were studied more than a century
ago by Hamel [16]. The problem was formulated rigorously in 1960’s by Rapcsák [27] and Klein
and Voutier [17]. Yet, the projective metrizability problem is far from being solved, and in the
last decade it has been intensively studied [1, 5, 10, 11, 29, 30, 31, 32].
3.1 Spray, nonlinear connection, and curvature
In this subsection, we start with a spray S and use the Frölicher–Nijenhuis theory to derive
a differential calculus on TM \ {0} [14] and to obtain information about the given system of
SODE. For the remaining part of the paper, all geometric objects will be considered defined
on the slashed tangent bundle TM \ {0} and not on the whole TM . This is motivated by the
fact that we will want to connect them with geometric structures in Finsler geometry, where the
Finsler function is not differentiable on the zero section.
Definition 3.1. A vector field S ∈ X(TM \ {0}) is called a spray if
i) JS = C,
ii) [C, S] = S.
First condition in Definition 3.1 expresses that a spray S can be locally given as
S = yi
∂
∂xi
− 2Gi(x, y)
∂
∂yi
,
for some functions Gi defined on domains of induced coordinates on TM \ {0}.
Second condition in Definition 3.1 expresses that the vector field S is 2-homogeneous. It is
equivalent with the fact that functions Gi are 2-homogeneous in the fibre coordinates. In this
work we will consider positive homogeneity only and hence Gi(x, λy) = λ2Gi(x, y) for all λ > 0.
By Euler’s theorem this homogeneity condition is equivalent to C(Gi) = 2Gi.
Projective Metrizability and Formal Integrability 9
A curve c : I → M is called regular if its tangent lift takes values in the slashed tangent
bundle, c′ : I → TM \ {0}. A regular curve is called a geodesic of spray S if S ◦ c′ = c′′. Locally,
c(t) = (xi(t)) is a geodesic of spray S if
d2xi
dt2
+ 2Gi
(
x,
dx
dt
)
= 0.
Definition 3.2. A nonlinear connection (or a horizontal distribution, or Ehresmann connection)
is defined by an n-dimensional distribution H : u ∈ TM \ {0} → Hu ⊂ Tu(TM \ {0}) that is
supplementary to the vertical distribution.
Every spray induces a nonlinear connection through the corresponding horizontal and vertical
projectors, [14]
h =
1
2
(Id−LSJ) , v =
1
2
(Id +LSJ) .
Locally, the two projectors h and v can be expressed as follows
h =
δ
δxi
⊗ dxi, v =
∂
∂yi
⊗ δyi,
where
δ
δxi
=
∂
∂xi
−N j
i (x, y)
∂
∂yj
, δyi = dyi +N i
j(x, y)dxj , N i
j(x, y) =
∂Gi
∂yj
(x, y).
Horizontal projector h is a vector valued almost semi-basic 1-form.
For a spray S consider the vector valued semi-basic 1-form
Φ = −v ◦ LSv = v ◦ LSh = v ◦ LS ◦ h,
which will be called the Jacobi endomorphism. It is also known as the Douglas tensor [15,
Definition 3.17] or as the Riemann curvature [29, Definition 8.1.2]. Locally, the Jacobi endo-
morphism can be expressed as follows
Φ = Rij(x, y)
∂
∂yi
⊗ dxj , Rij = 2
δGi
δxj
− S(N i
j) +N i
kN
k
j .
Another important geometric structure induced by a spray S is the curvature tensor R. It
is the vector valued semi-basic 2-form
R =
1
2
[h, h] =
1
2
Rijk
∂
∂yi
⊗ dxj ∧ dxk. (3.1)
Locally, the components of the curvature tensor, Rijk, are given by
Rijk =
δN i
j
δxk
−
δN i
k
δxj
.
Curvature tensor R expresses the obstruction to the integrability of the nonlinear connection.
Using formulae (2.4) and (3.1) we have that d2h = dR.
All the geometric objects induced by a spray S inherit the homogeneity condition. There-
fore [C, h] = 0, which means that the nonlinear connection is 1-homogeneous. Also [C, R] = 0,
[C,Φ] = Φ and hence the the curvature tensor R is 1-homogeneous, while the Jacobi endomor-
phism Φ is 2-homogeneous.
10 I. Bucataru and Z. Muzsnay
Using the Jacobi identity, [15, Proposition 2.7] , for the vector valued 0-form S and the
vector valued 1-form J we have [J, [S, J ]] − [J, [J, S]] − [S, [J, J ]] = 0. Therefore, we obtain
[J, h] = −2[J, [S, J ]] = 0.
The two semi-basic vector vector valued 1 and 2-forms Φ and R are related as follows:
Φ = iSR, [J,Φ] = 3R. (3.2)
First formula in (3.2) is a consequence of the homogeneity, while the second one is true in a more
general context. Locally, the above two formulae can be expressed as follows:
Rij = Rikjy
k, Rijk =
1
3
(
∂Rik
∂yj
−
∂Rij
∂yk
)
.
An important class of sprays, which we will use in the last section to provide examples of
projectively metrizable sprays, is that of isotropic sprays, [15, Definition 3.29].
Definition 3.3. A spray S is called isotropic if its Jacobi endomorphism has the form
Φ = λJ + η ⊗ C, (3.3)
where λ ∈ C∞(TM \ {0}) and η is a semi-basic 1-form on TM \ {0}.
Due to first formula in (3.2) we have that iSΦ = 0 and hence λ = −iSη. Also formulae (3.2)
allows us to express the isotropy condition (3.3) for a spray in terms of the curvature tensor R.
Proposition 3.4. A spray S is isotropic if and only if its curvature tensor R has the form
R = α ∧ J + β ⊗ C, (3.4)
where α is a semi-basic 1-form and β is a semi-basic 2-form on TM \ {0}.
Proof. We will prove that formulae (3.3) and (3.4) are equivalent.
Suppose that spray S is isotropic. Therefore, the Jacobi endomorphism Φ satisfies formu-
la (3.3). Using second formula (3.2), the formulae for the Frölicher–Nijenhuis bracket of two
vector valued forms [15, Appendix A1], and [J,C] = J , we have
3R = [J,Φ] = [J, λJ + η ⊗ C] = (dJλ− η) ∧ J + dJη ⊗ C.
Hence, the curvature tensor R has the form (3.4).
We assume now that the curvature tensor R has the form (3.4). Using first formula (3.2)
and the fact that the inner product iS is a derivation of degree −1, we have that the Jacobi
endomorphism has the form
Φ = iSR = iSαJ + (iSβ − α)⊗ C.
Hence, the spray S is isotropic. �
We will use Proposition 3.4 and formula (3.4) in Subsection 5.1 to show that isotropic sprays
are projectively metrizable sprays.
Projective Metrizability and Formal Integrability 11
3.2 Projectively related sprays
Two sprays are projectively equivalent if their geodesics coincide as oriented curves. Therefore,
a spray is called projectively metrizable if its geodesics coincide, as oriented curves, with the
geodesics of a Finsler space.
In [5] it has been shown that the Helmholtz conditions for an arbitrary semispray to be a Lag-
rangian vector field can be reformulated in terms of semi-basic 1-forms. It has been shown also
that out of the four classic Helmholtz conditions only two of them are necessary and sufficient in
the case of the projective metrizability problem for a spray. In this subsection we obtain directly
the two Helmholtz conditions, for projective metrizability, in terms of semi-basic 1-forms
Definition 3.5. By a Finsler function we mean a continuous function F : TM → R satisfying
the following conditions:
i) F is smooth on TM \ {0};
ii) F is positive on TM \ {0} and F (x, 0) = 0;
iii) F is positively homogeneous of order 1, which means that F (x, λy) = λF (x, y), for all
λ > 0 and (x, y) ∈ TM ;
iv) the metric tensor with components
gij(x, y) =
1
2
∂2F 2
∂yi∂yj
has rank n.
According to Lovas [21], conditions ii) and iv) of Definition 3.5 imply that the metric tensor gij
of a Finsler function is positive definite.
The regularity condition iv) of Definition 3.5 implies that the Euler–Poincaré 2-form of F 2,
ωF 2 = ddJF
2, is non-degenerate and hence it is a symplectic structure [20, 25]. Therefore, the
equation
iSddJF
2 = −dF 2 (3.5)
uniquely determine a vector field S on TM \ {0} that is called the geodesic spray of the Finsler
function. Equation (3.5) is equivalent to
LSdJF 2 = dF 2. (3.6)
Locally, the Euler–Poincaré 2-form of F 2, ωF 2 = ddJF
2, can be expressed as follows
ωF 2 = 2gijδy
i ∧ dxj .
Definition 3.6. A spray S is called Finsler metrizable if there exists a Finsler function F that
satisfies one of the two equivalent conditions (3.5) or (3.6).
One can reformulate condition iv) of Definition 3.5 in terms of the Hessian of the Finsler
function F as follows. Consider
hij(x, y) = F
∂2F
∂yi∂yj
the angular metric of the Finsler function. The metric tensor gij and the angular tensor hij are
related by
gij = hij +
∂F
∂yi
∂F
∂yj
.
Metric tensor gij has rank n if and only if angular tensor hij has rank (n−1), see [23]. Therefore,
the regularity of the Finsler function F is equivalent with the fact that the Euler–Poincaré 2-form
ωF = ddJF has rank 2n− 2.
12 I. Bucataru and Z. Muzsnay
Definition 3.7.
i) Two sprays S1 and S2 are projectively equivalent if their geodesics coincide up to an
orientation preserving reparameterization.
ii) A spray S is projectively metrizable if it is projectively equivalent to the geodesic spray of
a Finsler function.
Two sprays S1 and S2 are projectively equivalent if and only if there exists a 1-homogeneous
function P ∈ C∞(TM \ {0}) such that S2 = S1 − 2PC, [3, 29].
Next theorem gives a characterization of projectively metrizable sprays in terms of semi-basic
1-forms on TM \ {0}.
Theorem 3.8. A spray S is projectively metrizable if and only if there exists a semi-basic 1-form
θ ∈ Λ1
v(TM \ {0}) such that
rank (dθ) = 2n− 2, iSθ > 0, (3.7)
LCθ = 0, dJθ = 0, dhθ = 0. (3.8)
Proof. We prove first that conditions (3.7) and (3.8) are necessary for the projective metri-
zability problem of the spray S. We assume that S is projectively metrizable. Therefore,
there exists a Finsler function F with geodesic spray SF and a 1-homogeneous function P on
TM \{0} such that S = SF −2PC. Consider θ = dJF , the Euler–Poincaré 1-form of the Finsler
function F . Due to the 1-homogeneity condition of F it follows that iSθ = C(F ) = F > 0. The
non-degeneracy of the Finsler function implies rank (dθ) = 2n− 2. Since θ is 0-homogeneous it
follows that LCθ = 0. Condition dJθ = 0 is also satisfied since dJθ = d2JF = 0.
It remains to show that dhθ = 0. The geodesic spray SF is uniquely determined by condi-
tion (3.5), from which it follows that SF (F 2) = 0 and hence SF (F ) = 0. Since SF also satisfies
condition (3.6) it follows that LSF (Fθ) = FdF , which implies LSF θ = dF . Using S = SF −2PC
we obtain that LSθ − 2LPCθ = dF . Using again the 0-homogeneity of the semi-basic 1-form θ
it follows LPCθ = PLCθ = 0 and hence LSθ = dF . We apply now dJ to both sides of this last
relation and use the commutation rules LSdJ − dJLS = d[S,J ] = −dh + dv and ddJ + dJd = 0.
Therefore,
−dhθ − dvθ = −ddJF = dJdF = dJLSθ = LSdJθ + dhθ − dvθ,
from where it follows that dhθ = 0.
We prove now that the conditions (3.7) and (3.8) are sufficient for the projective metriz-
ability problem of the spray S. Consider θ ∈ Λ1(TM \ {0}) a semi-basic 1-form that sa-
tisfies conditions (3.7) and (3.8). Define the function F = iSθ. Using the commutation
rule iSdJ + dJ iS = LC − i[S,J ] as well as conditions dJθ = 0 and LCθ = 0 it follows that
dJF = dJ iSθ = ihθ = θ. Hence θ is the Euler–Poincaré 1-form of F . Now conditions (3.7) assure
that F is a Finsler function. Consider the function P ∈ C∞(TM \ {0}) given by 2P = S(F )/F ,
which is 1-homogeneous. We will show now that the spray S̃ = S− 2PC satisfies equation (3.6)
and hence it is the geodesic spray of the Finsler function F .
Using the commutation rule iSdh + dhiS = LS − i[S,h] and the fact that dhθ = 0 it follows
0 = iSdhθ = −dhiSθ+LSθ− i[S,h]θ. Using the fact that i[S,h]θ = dF ◦J ◦LSh = dF ◦ v = dvF it
follows that LSθ = dhF + dvF = dF . We show now that S̃ satisfies the same equation. Indeed
LS̃θ = LS−2PCθ = dF since LPCθ = 0. From the defining formula of function P is follows that
S̃(F ) = S(F ) − 2PC(F ) = S(F ) − 2PF = 0. Therefore LS̃dJF
2 = 2FLS̃dJF = 2FdF = dF 2
and hence S̃ is the geodesic spray of the Finsler function F . �
Projective Metrizability and Formal Integrability 13
The second part of the proof of Theorem 3.8 shows that if there exists a semi-basic 1-form θ
on TM \ {0} that satisfies the conditions (3.7) and (3.8) then the given spray S is projectively
related to the spray
SF = S − LS(iSθ)
iSθ
C,
which is the geodesic spray of the Finsler function F = iSθ. In this case, the semi-basic 1-form
θ = θidx
i is the Euler–Poincaré 1-form of the Finsler function F , θ = dJF . Therefore,
θi =
∂F
∂yi
, hij = F
∂θi
∂yj
, Fdθ = hijδy
i ∧ dxj . (3.9)
Formulae (3.9) show the relation between a semi-basic 1-form θ, a solution of the projective
metrizability problem using Theorem 3.8, and the classic approach of the problem using the
multiplier matrix hij .
4 Formal integrability for the projective metrizability problem
Theorem 3.8 provides necessary and sufficient conditions for the projective metrizability prob-
lem. These conditions consist of a set of algebraic equations (3.7), and a set of differential
equations (3.8). In this section, we study the set of differential equations (3.8) using Spencer’s
technique of formal integrability [7, 15] and following some of the techniques used for studying
the Finsler metrizability problem, which were developed in [26].
4.1 Formal integrability
In this subsection, we recall first the basic notions of formal integrability [7, 15] and then we
apply it to the system (3.8).
Consider E a vector bundle over the base manifold M . For a section s of E and k ≥ 1
we denote by jkxs the kth order jet of s at the base point x in M . The bundle of kth order
jets of sections of E is denoted by JkE. For two vector bundles E and F over the same base
manifold M , a linear partial differential operator of order k,
P : Sec(E)→ Sec(F ),
can be identified with a morphism of vector bundles over M , p0(P ) : JkE → F . We will also
consider the lth order jet prolongation of the differential operator P , which will be identified
with the morphisms of vector bundles over M , pl(P ) : Jk+lE → J lF , defined by
pl(P )
(
jk+lx s
)
= jlx(Ps).
We will denote by Rk+lx (P ) = Ker plx(P ) ⊂ Jk+lx E the space of (k + l)th order formal solutions
of P at x in M .
Definition 4.1. The differential operator P is called formally integrable at x in M if Rk+l(P )
is a vector bundle over M , for all l ≥ 0, and the map π̄k+l−1x : Rk+lx (P )→ Rk+l−1x (P ) is onto for
all l ≥ 1.
In the analytic case, formal integrability implies existence of analytic solutions for arbitrary
initial data, see [7, p. 397].
Denote by σk(P ) : Sk(M) ⊗ E → F the symbol of P , which is defined by the highest order
terms of the differential operator P , and by σk+l(P ) : Sk+l(M) ⊗ E → Sl(M) ⊗ F the symbol
of the lth order prolongation of P . For each x in M , we write
gkx(P ) = Kerσkx(P ),
14 I. Bucataru and Z. Muzsnay
gkx(P )e1...ej = {A ∈ gkx(P )|ie1A = · · · = iejA = 0}, j ∈ {1, . . . , n},
where {e1, . . . , en} is a basis of TxM . Such a basis is called quasi-regular if it satisfies
dim gk+1
x (P ) = dim gkx(P ) +
n∑
j=1
dim gkx(P )e1...ej . (4.1)
The symbol σk(P ) is called involutive at x in M if there exists a quasi-regular basis of TxM .
In this work we will address the projective metrizability problem by discussing first the formal
integrability of the system (3.8). For this we will use the two sufficient conditions provided by
Cartan–Kähler theorem.
Theorem [Cartan–Kähler]. Let P be a linear partial differential operator of order k. Suppose
gk+1(P ) is a vector bundle over Rk(P ). If the map πk : Rk+1(P ) → Rk(P ) is onto and the
symbol σk(P ) is involutive, then P is formally integrable.
In order to study the formal integrability of the system (3.8) we consider the first-order partial
differential operator P1 : Λ1
v(TM \{0})→ Λ1
v(TM \{0})⊕Λ2
v(TM \{0})⊕Λ2
v(TM \{0}), which
we call the projective metrizability operator
P1 = (LC, dJ , dh) . (4.2)
Since C and J are vector valued, semi-basic 0 and respectively 1-forms and h is a vector valued
almost semi-basic 1-form, according to Lemma 2.4, all differential operators LC, dJ , dh preserve
semi-basic forms. Therefore, the differential operator P1 is well defined.
4.2 Involutivity of the projective metrizability operator
In this subsection we prove that the projective metrizability operator (4.2) satisfies one of the
two sufficient conditions for formal integrability, provided by Cartan–Kähler theorem, namely
we will prove that the symbol σ1(P1) is involutive.
Since all the bundles we will refer to in this subsection are vector bundles over TM \ {0},
we will omit mentioning it explicitly. For example, we will denote by T ∗v the vector bundle
of semi-basic 1-forms T ∗v (TM \ {0}), which is a subbundle of T ∗(TM \ {0}). We will denote
by ΛkT ∗v the vector bundle of semi-basic k-forms on TM \ {0}, and by Λkv = Sec
(
ΛkT ∗v
)
the
C∞(TM \ {0})-module of sections Λkv(TM \ {0}). By SkT ∗ we denote the vector bundle of
symmetric tensors of (0, k)-type on TM \ {0}.
The partial differential operator P1 induces a morphism of vector bundles
p0(P1) : J1T ∗v → F1 := T ∗v ⊕ Λ2T ∗v ⊕ Λ2T ∗v .
Together with this morphism we will consider the lth order jet prolongations pl(P1) : J l+1T ∗v →
J lF1, for l ≥ 1.
Locally, for a semi-basic 1-form θ = θidx
i ∈ Λ1
v, we have
LCθ =
∂θi
∂yj
yjdxi, dJθ =
1
2
(
∂θi
∂yj
− ∂θj
∂yi
)
dxj ∧ dxi,
dhθ =
1
2
(
δθi
δxj
− δθj
δxi
)
dxj ∧ dxi.
Therefore, the vector bundle morphism p0(P1) can be expressed as follows
p0(P1)(j
1θ) =
(
∂θi
∂yj
yjdxi,
1
2
(
∂θi
∂yj
− ∂θj
∂yi
)
dxj ∧ dxi, 1
2
(
δθi
δxj
− δθj
δxi
)
dxj ∧ dxi
)
.
Projective Metrizability and Formal Integrability 15
The symbol of P1 is the vector bundle morphism σ1(P1) : T ∗⊗T ∗v → F1, defined by the highest
order terms of p0(P1). Since all terms that define p0(P1) are first-order terms, it follows that
σ1(P1)A =
(
σ1 (LC)A = iCA, σ
1 (dJ)A = τJA, σ
1 (dh)A = τhA
)
. (4.3)
In view of formula (2.1), the three components of the vector bundle morphism σ1(P1) are
given by:(
σ1 (LC)A
)
(X) = (iCA) (X) = A(C, X);(
σ1 (dJ)A
)
(X,Y ) = (τJA) (X,Y ) = A(JX, Y )−A(JY,X);(
σ1 (dh)A
)
(X,Y ) = (τhA) (X,Y ) = A(hX, Y )−A(hY,X),
for X, Y vector fields on TM \ {0}. Note that for A ∈ T ∗ ⊗ T ∗v , iCA, τJA, τhA are semi-basic
forms and hence the symbol σ1(P1) is well defined.
The first-order prolongation of the symbol of P1 is the vector bundle morphism σ2(P1) :
S2T ∗⊗ T ∗v → T ∗⊗F1 that satisfies iX
(
σ2(P1)B
)
= σ1(P1)(iXB) for all B ∈ S2T ∗⊗ T ∗v and all
X ∈ X(TM \ {0}). Therefore, we obtain
σ2(P1)B =
(
σ2 (LC)B, σ2 (dJ)B, σ2 (dh)B
)
,
where for X,Y, Z vector fields on TM \ {0} we have:(
σ2 (LC)B
)
(X,Y ) = B(X,C, Y ),(
σ2 (dJ)B
)
(X,Y, Z) = B(X, JY, Z)−B(X, JZ, Y ), (4.4)(
σ2 (dh)B
)
(X,Y, Z) = B(X,hY, Z)−B(X,hZ, Y ).
Theorem 4.2. The symbol σ1(P1), of the projective metrizability operator P1 = (LC, dJ , dh), is
involutive.
Proof. The symbol σ1(P1) is involutive if there exists a quasi-regular basis of Tu(TM \ {0}).
It means that we will have to seek for a basis of Tu(TM \ {0}) that satisfies the equality (4.1)
for k = 1, at some point u ∈ TM \ {0}.
We start by computing the first term in the right hand side of formula (4.1), which is
dim g1u(P1), for some u ∈ TM \ {0}. Recall that g1(P1) = Ker
(
σ1(P1)
)
⊂ T ∗ ⊗ T ∗v . We
have to compute the dimension of the fibers of g1(P1), which is a vector subbundle of T ∗ ⊗ T ∗v .
An element A ∈ g1(P1) can be expressed, with respect to the adapted dual basis {dxi, δyi}, as
follows
A = Aijdx
i ⊗ dxj +Aijδy
i ⊗ dxj .
Using formula (4.3), the symbol σ1(P1) can be expressed as follows:
σ1(P1)A =
(
Aijy
idxj ,
1
2
(Aij −Aji)dxi ∧ dxj ,
1
2
(Aij −Aji)dxi ∧ dxj
)
.
The condition τhA = 0 is equivalent with Aij = Aji and due to this condition Aij contribute
with n(n + 1)/2 to the dim g1u(P1). The conditions τJA = 0 and iCA = 0 are equivalent to
Aij = Aji, and respectively Aijy
i = 0. Hence, due to these two conditions, Aij contribute with
n(n− 1)/2 to the the dim g1u(P1). It follows that dim g1u(P1) = n(n− 1)/2 + n(n+ 1)/2 = n2.
We continue the proof by computing the left hand side of formula (4.1), which is dim g2u(P1).
Therefore, we will consider the kernel of the first-order prolongation of the symbol, g2(P1) =
16 I. Bucataru and Z. Muzsnay
Ker
(
σ2(P1)
)
⊂ S2T ∗ ⊗ T ∗v . An element B ∈ g2(P1) can be expressed, with respect to the
adapted dual basis {dxi, δyi}, as follows
B = Bijkdx
i ⊗ dxj ⊗ dxk +Bijkδy
i ⊗ dxj ⊗ dxk
+Bijkdx
i ⊗ δyj ⊗ dxk +Bijkδy
i ⊗ δyj ⊗ dxk,
with the symmetry conditions Bijk = Bjik, Bijk = Bijk and Bijk = Bjik satisfied. Using
formula (4.4), the symbol σ2(P1) can be expressed as follows
σ2(P1)B =
(
Bijky
jdxi ⊗ dxk +Bijky
jδyi ⊗ dxk,
1
2
(Bijk −Bikj)dxi ⊗ dxj ∧ dxk +
1
2
(Bijk −Bikj)δyi ⊗ dxj ∧ dxk,
1
2
(Bijk −Bikj)dxi ⊗ dxj ∧ dxk +
1
2
(Bijk −Bikj)δyi ⊗ dxj ∧ dxk
)
.
The totally symmetric components Bijk contribute with n(n + 1)(n + 2)/6 to the dim g2u(P1).
The other two are also totally symmetric components on the (n−1)-dimensional space given by
restrictions Bijky
j = 0 and respectively Bijky
j = 0. Therefore, each of them contributes with
(n − 1)n(n + 1)/6 to the dim g2u(P1). Consequently, dim g2u(P1) = n(n + 1)(n + 2)/6 + 2(n −
1)n(n+ 1)/6 = n2(n+ 1)/2.
Finally, for some u ∈ TM \ {0}, we seek for a basis of Tu(TM \ {0}) for which formula (4.1)
holds true. Consider {hi, i ∈ {1, . . . , n}} a basis for the horizontal distribution and {vi, i ∈
{1, . . . , n}}, with vn = C, a basis for the vertical distribution such that Jhi = vi, for all
i ∈ {1, . . . , n}.
For A ∈ g1(P1), and the basis B = {hi, vi, i ∈ {1, . . . , n}}, let us denote
aij = A(hi, hj) and bij = A(hi, vj).
It follows that
1. aij = aji, i, j = 1, . . . , n, because A ∈ Kerσ1(dh),
2. bij = bji, i, j = 1, . . . , n, because A ∈ Kerσ1(dJ),
3. bni = (bin) = 0, i = 1, . . . , n, because A ∈ Kerσ1(LC).
Note that dim g1(P1) = n2 is determined by the n(n+ 1)/2 independent components of aij and
n(n− 1)/2 independent components of bij . We will prove now that B̃ = {ei, vi, i ∈ {1, . . . , n}},
where e1 = h1, e2 = h2 + v1, . . . , en−1 = hn−1 + vn−2, en = S + vn−1, is a quasi-regular basis.
For the basis B̃ we denote
ãij = A(ei, ej) and b̃ij = A(vi, ej).
Because A is semi-basic in the second variable we have
A(ei, ej) = A(hi + vi−1, hj + vj−1) = A(hi + vi−1, hj),
A(vi, ej) = A(vi, hj + vj−1) = A(vi, hj),
which means that
ãij = aij + bi−1,j , b̃ij = bij , i, j = 1, . . . , n.
Moreover, the n2 independent components aij and bij of A in the basis B can be obtained from
the components ãij in the basis B̃.
Projective Metrizability and Formal Integrability 17
Now, for each j ∈ {1, . . . , n} we have that conditions ie1...ejA = 0 give jn independent
restrictions on the n2-dimensional space g1u(P1). This implies that
dim g1u(P1)e1...ej = n(n− j), dim g1u(P1)e1...en,v1,...vj = 0.
It follows that
dim g1u (P1) +
n∑
i=1
dim g1u (P1)e1,...,ei +
n∑
i=1
dim g1u (P1)e1,...,en,v1,...,vi
= n2 + n(n− 1) + · · ·+ n = n2(n+ 1)/2 = dim g2u (P1) ,
which shows that formula (4.1) is satisfied for P1, k = 1, and the basis B̃. Therefore, B̃ is
a quasi-regular basis and hence the symbol of P1 is involutive. �
4.3 First obstruction for the projective metrizability problem
We have seen in the previous subsection that one condition, of the two sufficient conditions of
the Cartan–Kähler theorem, for the formal integrability of P1, is satisfied. In this subsection
we address the second sufficient condition. We prove that there is only one obstruction for
the formal integrability of the projective metrizability operator P1 and this is due to the the
curvature tensor R of the induced nonlinear connection.
Theorem 4.3. A first-order formal solution θ ∈ Λ1
v of the system (3.8) can be lifted into a
second-order solution, which means that π1 : R2(P1)→ R1(P1) is onto, if and only if
dRθ = 0, (4.5)
where R is the curvature tensor (3.1).
Proof. Using the notations from Subsection 4.2, we denote by K, the cokernel of the mor-
phism σ2(P1),
K =
T ∗ ⊗
(
T ∗v ⊕ Λ2T ∗v ⊕ Λ2T ∗v
)
Imσ2(P1)
. (4.6)
We will prove the theorem by using the following classical result of homological algebra, see [15,
Proposition 1.1]. There exists a morphism ϕ : R1(P1)→ K such that the sequence
R2(P1)
π1−→ R1(P1)
ϕ−→ K
is exact. In particular, the morphism π1 is onto if and only if ϕ = 0.
We will build the morphism ϕ and show that for θ ∈ Λ1
v such that j1uθ ∈ R1
u(P1) ⊂ J1
uT
∗
v ,
a first-order solution of P1 at u ∈ TM \ {0} we have that ϕuθ = 0 if and only if (dRθ)u = 0.
The morphism ϕ is represented in the diagram (4.7) by dashed arrows.
To build ϕ, we have to define first a morphism of vector bundles
τ : T ∗ ⊗
(
T ∗v ⊕ Λ2T ∗v ⊕ Λ2T ∗v
)
→ K,
18 I. Bucataru and Z. Muzsnay
such that the first row in the following diagram is exact.
0
��
0
��
0
��
0 // g2(P1) //
��
S2T ∗ ⊗ T ∗v
σ2(P1) //
ε
��
T ∗ ⊗ F1
////___τ
ε
��
K // 0
0 // R2(P1)
i //
π1
��
J2T
∗
v
//
π1
��
//_____ p1(P1)
J1F1
π
��
OO�
�
�∇
0 // R1(P1) // //____ i
J1T ∗v
po(P1) //
OO�
�
�
��
F1
��
0 0
(4.7)
For the vector bundle K given by formula (4.6), the dimension of its fibres is n2(n − 1)/2.
Therefore, we can view this vector bundle over TM \ {0} as follows
K = ⊕(2)Λ2T ∗v ⊕(3) Λ3T ∗v .
Therefore the map τ has 5 components. The five components of τ = (τ1, . . . , τ5), are given as
follows
τ1(A,B1, B2) = τJA− iCB1, τ2(A,B1, B2) = τhA− iCB2, τ3(A,B1, B2) = τJB1,
τ4(A,B1, B2) = τhB2, τ5(A,B1, B2) = τhB1 + τJB2,
for A ∈ T ∗ ⊗ T ∗v , B1, B2 ∈ T ∗ ⊗ Λ2T ∗v . Using the above definition of the five components of τ ,
formula (4.4) that defines the three components of σ2(P1), and the symmetry in the first two
arguments of an element B ∈ S2T ∗ ⊗ T ∗v we can prove
(
τ ◦ σ2(P1)
)
(B) = 0. For example, the
first component of this composition is given by(
τ1 ◦ σ2(P1)
)
(B)(X,Y ) =
(
τJσ
2(LC)
)
(B)(X,Y )−
(
iCσ
2(dJ)
)
(B)(X,Y )
= B(JX,C, Y )−B(JY,C, X)−B(C, JX, Y ) +B(C, JY,X) = 0.
It follows that Imσ2(P1) ⊂ Ker τ . By comparing the dimensions, it is easy to see that
Imσ2(P1) = Ker τ , and therefore the first row in diagram (4.7) is exact.
Consider ∇ a linear connection on TM \ {0} such that ∇J = 0. It follows that the connec-
tion ∇ preserves the vertical distribution and hence it will preserve semi-basic forms. Therefore,
one can view ∇ as a connection on the fibre bundle F1 → TM \{0}. Using Lemma 2.5, it follows
that derivations DC = iC∇, DJ = τJ∇, and Dh = τh∇ preserve semi-basic forms. As a first-
order partial differential operator, we can identify connection ∇ with the bundle morphism
p0(∇) : J1F1 → T ∗ ⊗ F1. We will use this bundle morphism to define the map ϕ : R1(P1)→ K
we mentioned at the beginning of the proof.
Consider θ ∈ Λ1
v such that j1uθ ∈ R1
u(P1) ⊂ J1
uT
∗
v is a first-order solution of P1 at u ∈ TM\{0}.
Then, we define
ϕuθ = τu∇P1θ = τu(∇LCθ,∇dJθ,∇dhθ).
We will compute now the five components of map ϕ. Since LCθ, dJθ, and dhθ vanish at
u ∈ TM \ {0}, using Lemma 2.1, it follows that when acting on this semi-basic forms we have
Projective Metrizability and Formal Integrability 19
DC = LC, DJ = dJ , and Dh = dh. Using the fact that [J,C] = J , [h,C] = 0, [J, J ] = 0, and
[h, J ] = 0, it follows that
τ1 (∇P1θ)u = (τJ∇LCθ − iC∇dJθ)u = (dJLCθ − LCdJθ)u = (d[J,C]θ)u = 0;
τ2 (∇P1θ)u = (τh∇LCθ − iC∇dhθ)u = (dhLCθ − LCdhθ)u = (d[h,C]θ)u = 0;
τ3 (∇P1θ)u = (τJ∇dJθ)u =
(
d2Jθ
)
u
=
1
2
(d[J,J ]θ)u = 0;
τ4 (∇P1θ)u = (τh∇dhθ)u =
(
d2hθ
)
u
=
1
2
(d[h,h]θ)u = (dRθ)u;
τ5 (∇P1θ)u = (τh∇dJθ + τJ∇dhθ)u = (d[h,J ]θ)u = 0.
From the above calculations it follows that a first-order formal solution θ of the system (3.8)
can be lifted into a second-order solution if and only if dRθ = 0. �
Using notation (3.9) we can rewrite obstruction condition (4.5) as an algebraic Bianchi iden-
tity for the curvature tensor
FiRdθ = hikR
k
jl + hlkR
k
ij + hjkR
k
li = 0. (4.8)
An alternative expression for the algebraic Bianchi identity (4.8) was obtained by Szilasi and
Vattamány in [31, 4.9.1a].
Using formula (3.1), we obtain that any solution of the system (3.8) necessarily satisfies
the curvature obstruction (4.5). In the next section we will discuss various cases when the
obstruction (4.5) is automatically satisfied. Another possibility, which we leave for further
work, is to add this obstruction to the projective metrizability operator P1. In this case we
can consider the first-order partial differential operator P2 : Λ1
v(TM \ {0})→ Λ1
v(TM \ {0})⊕
Λ2
v(TM \ {0})⊕ Λ2
v(TM \ {0})⊕ Λ3
v(TM \ {0}),
P2 = (LC, dJ , dh, dR) .
Following a similar approach as we did for the projective metrizability operator P1, we can use
the Cartan–Kähler theorem to study the formal integrability of the differential operator P2.
5 Classes of sprays that are projectively metrizable
In this section we present three classes of sprays for which the projective metrizability opera-
tor P1 is formally integrable and hence the system (3.8) always has solutions. Therefore, for
each of these classes we address the projective metrizability problem, by discussing the set of
algebraic conditions (3.7) only, which as we show are always satisfied. We will also provide
examples of projectively metrizable sprays that are not Finsler metrizable.
5.1 Projectively metrizable sprays
In this subsection we assume that a spray S is analytic, on an analytic manifold M . We show
that if for spray S the projective metrizability operator P1 is formally integrable then the spray
is projectively metrizable.
For a semi-basic 1-form θ = θi(x, y)dxi ∈ Λ1
v, we will express its first-order jet j1θ ∈ J1T ∗v in
the adapted dual basis {dxi, δyi}, induced by the nonlinear connection associated to the spray,
which means
j1θ =
δθi
δxj
dxj ⊗ dxi +
∂θi
∂yj
δyj ⊗ dxi.
20 I. Bucataru and Z. Muzsnay
This expression provides us local coordinates (xi, yi, θi, θij , θij) for J1T ∗v . The typical fibre for
the fibre bundle J1T ∗v → TM \ {0} is Rn∗ × L2(n,R) × L2(n,R). With respect to these local
coordinates, the fibre R1
u(P1) of first-order formal solution of P1 at u = (xi, yi) ∈ TM \ {0} can
be expressed as follows
R1
u(P1) = {(xi, yi, θi, θij , θij) ∈ J1
uT
∗
v , θij = θji, θij = θji, θijy
j = 0}.
Hence the typical fibre of the fibre bundle R1(P1)→ TM \{0} is Rn∗×L2,s(n,R)×L2,s(n−1,R),
where L2,s(n,R) is the space of bilinear symmetric forms on Rn.
Consider θ a solution of the system (3.8), with the initial data (θ0i , θ
0
ij , θ
0
ij) ∈ R1
u(P1) satisfying
the algebraic conditions (3.7). This means θ0i y
i > 0 (in each fibre, yi is a fixed direction, hence
one can choose (θ0i ) ∈ Rn∗ such that θ0i y
i > 0) and rank(θ0ij) = n−1 (choose (θ0ij) ∈ L2,s(n−1,R)
and extend it to Rn−1 ⊕ {a(yi), a ∈ R} such that θ0ijy
j = 0). If we assume that M is connected
and dimM ≥ 2, then TM \ {0} is also connected. Therefore, due to continuity, the solution θ
satisfies the algebraic conditions (3.7), on the connected component of u ∈ TM \ {0} where θ is
defined.
We present now some classes of sprays for which the projective metrizability operator P1 is
always integrable, and hence these sprays will be projectively metrizable. These classes of sprays
are:
i) flat sprays, R = 0;
ii) isotropic sprays, R = α ∧ J + β ⊗ C, for α a semi-basic 1-form and β a semi-basic 2-form
on TM \ {0};
iii) arbitrary sprays on 2-dimensional manifolds.
For each of these classes of sprays, we will show that the curvature obstruction is automati-
cally satisfied and hence the projective metrizability problem will always have a solution in the
analytic case.
In the flat case, the obstruction is automatically satisfied. The fact that flat sprays are
projectively metrizable was already demonstrated with other methods in [9].
Assume that a spray S is isotropic. It follows that the curvature tensor has the form R =
α ∧ J + β ⊗ C, for α ∈ Λ1
v and β ∈ Λ2
v. Then, for a semi-basic 1-form θ on TM \ {0}, we have
dRθ = α ∧ dJθ + β ⊗ LCθ. (5.1)
If θ is a solution of the differential system (3.8) it follows that LCθ = 0 and dJθ = 0, and using
formula (5.1) it follows that dRθ = 0. Therefore, the obstruction for the formal integrability
of P1 is satisfied. In [10] it has been shown that any isotropic sprays is projectively equivalent
to a flat spray and hence it is projectively metrizable.
If dimM = 2 then for a semi-basic 1-form θ on TM \ {0}, dRθ is a semi-basic 3-form and
hence it will have to vanish. It has been shown by Matsumoto [22] that every spray on a surface
is projectively related to a Finsler space, using the original discussion of Darboux [12] about
second-order differential equations.
5.2 Examples
In this subsection we provide examples of non-metrizable Finsler sprays in the last two of the
above mentioned classes of projectively metrizable sprays.
Consider the following system of second-order ordinary differential equations in some open
domain in R2, which was proposed by Anderson and Thompson in [2, Example 7.2]:
d2x1
dt2
+
(
dx1
dt
)2
+
(
dx2
dt
)2
= 0,
d2x2
dt2
+ 4
dx1
dt
dx2
dt
= 0. (5.2)
Projective Metrizability and Formal Integrability 21
The corresponding spray is
S = y1
∂
∂x1
+ y2
∂
∂x2
−
(
(y1)2 + (y2)2
) ∂
∂y1
− 4y1y2
∂
∂y2
. (5.3)
It has been shown in [2] that the system (5.2) is not variational and therefore the corresponding
spray S in formula (5.3) is not Finsler metrizable. We can also use the techniques from [26]
to show that the spray S in formula (5.3) is not Finsler metrizable. However, according to the
discussion in the previous subsection, the spray S is projectively metrizable.
Next we consider another example of projectively metrizable spray that is not Finsler metriz-
able, which was proposed by G. Yang in [32]. Consider F a projectively flat Finsler function on
some open domain U ⊂ Rn [29, § 13.5]. This means that the geodesic spray S of F is projectively
equivalent to a flat spray. Therefore, spray S is locally given by:
S = yi
∂
∂xi
− 2P (x, y)yi
∂
∂yi
,
where P is 1-homogeneous function on U × (Rn \ {0}). We assume that for the projectively flat
Finsler function F, its flag curvature is constant κ ∈ R, κ 6= 0, [15, § 3.5], [29, § 11.1]. This is
equivalent to the fact that the Jacobi endomorphism induced by the spray S has the form
Φ = κ
(
F 2J − FdJF ⊗ C
)
.
Yang shows in [32, Theorem 1.2] that the projective metrizability class of S contains sprays
that are not Finsler metrizable. More precisely, he shows that for λ ∈ R such that λ 6= 0 and
κ+ λ2 6= 0, then the spray
S̃ = S − 2λFC (5.4)
cannot be projectively flat and hence it is not Finsler metrizable.
For spray S̃ one can compute the corresponding geometric structures: nonlinear connection,
Jacobi endomorphism, curvature tensor in terms of the corresponding ones induced by spray S:
h̃ = h+ [PC, J ], Φ̃ = Φ + λ2(F 2J − FdJF ⊗ C), R̃ = R+ λ2FdJF ∧ J. (5.5)
Therefore S̃ has constant flag curvature κ + λ2 and it is also isotropic. Then one can also use
formulae (5.5) and Theorem 2 from [26], or Theorem 7.2 from [15], to show that Yang’s example
given in formula (5.4) is not Finsler metrizable. Yang’s example can be extended and it can
be shown that for an arbitrary spray, its projective class contains sprays that are not Finsler
metrizable, [6].
Therefore, spray S̃ in formula (5.4) is projectively metrizable but it is not Finsler metrizable.
Acknowledgements
The work of IB was supported by the Romanian National Authority for Scientific Research,
CNCS UEFISCDI, project number PN-II-RU-TE-2011-3-0017. The work of Z.M. has been
supported by the Hungarian Scientific Research Fund (OTKA) Grant K67617.
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http://dx.doi.org/10.1142/S0219887811005701
http://arxiv.org/abs/1011.5799
http://dx.doi.org/10.3934/jgm.2009.1.159
http://arxiv.org/abs/0903.1169
http://arxiv.org/abs/1108.4628
http://dx.doi.org/10.1088/0305-4470/14/10/012
http://dx.doi.org/10.1142/9789812813596
http://dx.doi.org/10.1007/BF01444348
http://dx.doi.org/10.1016/B978-044452833-9.50017-6
http://dx.doi.org/10.1007/BF02228993
http://dx.doi.org/10.1007/BF02228993
http://dx.doi.org/10.1016/0370-1573(90)90137-Q
http://arxiv.org/abs/math.DG/0602383
http://dx.doi.org/10.1088/0305-4470/15/5/013
http://dx.doi.org/10.1142/9789812790613_0045
http://dx.doi.org/10.1142/9789812790613_0045
http://dx.doi.org/10.1023/A:1014928103275
http://dx.doi.org/10.1016/j.difgeo.2011.04.041
1 Introduction
2 Preliminaries
2.1 Frölicher-Nijenhuis theory
2.2 Vertical calculus on TM and semi-basic forms
3 Projective metrizability problem of a spray
3.1 Spray, nonlinear connection, and curvature
3.2 Projectively related sprays
4 Formal integrability for the projective metrizability problem
4.1 Formal integrability
4.2 Involutivity of the projective metrizability operator
4.3 First obstruction for the projective metrizability problem
5 Classes of sprays that are projectively metrizable
5.1 Projectively metrizable sprays
5.2 Examples
References
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