PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 153–163 DOI: 10.2298/PIM150203020C

SEMI-BASIC 1-FORMS AND COURANT STRUCTURE FOR METRIZABILITY PROBLEMS

Mircea Crasmareanu

Abstract. The metrizability of sprays, particularly symmetric linear connections, is studied in terms of semi-basic 1-forms using the tools developed by Bucataru and Dahl in [2]. We introduce a type of metrizability in relationship with the Finsler and projective metrizability. The Lagrangian corresponding to the Finsler metrizability, as well as the Bucataru–Dahl characterization of Finsler and projective metrizability are expressed by means of the Courant structure on the big tangent bundle of T M. A byproduct of our computations is that a flat Riemannian metric, or generally an R-flat Finslerian spray, yields two complementary, but not orthogonally, Dirac structures on T^{big} T M. These Dirac structures are also Lagrangian subbundles with respect to the natural almost symplectic structure of T^{big} T M.

1. Introduction

The question whether a given symmetric linear connection ∇ on the manifold M is the Levi-Civita connection of a Riemannian metric is very important from both mathematical (where it appears as a reciprocal of the fundamental theorem of Riemannian geometry) and physical reasons [15]. So, it has a rich history (which is possible to begin with [9]) pointed out for example in [8], where the conditions for local metrizability as well as the global problem are discussed in detail. Recently, there is a growing interest for this question in control theory [12] and more generally, under the name of inverse problem, in geometrization of general and special Lagrangian systems [14, 16].

Here we introduce a type of metrizability which involves the geometry of the tangent bundle. Namely, we search for a metric g not on the base manifold M but on the tangent manifold T M by imposing that g is covariant constant with respect

2010 Mathematics Subject Classification: Primary 53C60; Secondary 53C05; 53C15; 53C20; 53C55; 53D18.

Key words and phrases: (semi)spray; metrizability; semi-basic 1-form; symplectic form; homogeneity; big tangent bundle; Courant bracket; Dirac structure; isotropic subbundle; Poincaré– Cartan 1-form.

Communicated by Zoran Rakić.

to the dynamical covariant derivative of the spray associated to ∇. A condition for this kind of metrizability is then derived by using the Finsler metrizability of sprays characterized by Bucataru and Dahl in [2] in terms of semi-basic 1-forms. The initial metrizability problem is recovered in terms of global smoothness of this 1-form and also there are considered several relationships with the symplectic geometry of T M induced by a 2-homogeneous Lagrangian L of Finsler type. In fact, the possibility to use the symplectic structures was the motivation for our choice of [2] as the main point of beginning the present study instead of classical holonomy methods, [17] or [19].

We express the above Finsler (and also the projective) metrizability of a general spray S and the corresponding Lagrangian L by means of the Courant structure on the big tangent bundle of T M. Essential terms in the Courant bracket for several objects are based on the coefficients of the Jacobi operator of S, as well as on the coefficients of the curvature of the nonlinear connection induced by S. For example, even in the Riemannian case we derive new results: the flatness of the Riemannian metric is equivalent with the Courant isotropy of the pair (the horizontal distribution, vertical dual of the Poincaré–Cartan 1-form). The same flatness yields a half-Dirac structure on T^{big}T M while the above subbundle together with (the vertical distribution, the Poincaré–Cartan 1-form) are isotropic with respect to the natural almost symplectic structure of the big tangent bundle of T M.

Moreover, we obtain a Dirac structure V_{D}(S)=(the vertical distribution, its annihilator) for every Lagrangian spray S i.e., S gives the Euler–Lagrange equations of a regular Lagrangian L; the regularity means that the Hessian of L(x, y) with respect to the vertical variables y is nondegenerate. With respect to the dual point of view, the pair H_{D}(S)=(the horizontal distribution, its annihilator) is a Dirac structure on the T^{big}T M if and only if the Finslerian spray S is R-flat which means the vanishing of the curvature of its nonlinear connection; this means in the particular case of Riemannian geometry that the metric is flat. In conclusion, for a flat Riemannnian metric the big tangent bundle of T M is the complementary (but not orthogonally) sum of two Dirac structures, both being also Lagrangian subbundles for the almost symplectic structure of T^{big}T M as well as Lie algebroids over T M. Let us remark that some classes of Dirac structures naturally associated to Lagrangian systems appear also in [20] and the above results are already obtained for the Riemannian case in [8].

2. Semibasic 1-forms for the metrizability of linear connections

Fix M a smooth n ⩾ 2 dimensional manifold with the tangent bundle T M and cotangent bundle T^{∗}M. Local coordinates on M will be denoted by x = (x^{i}), 1 ⩽ i ⩽ n while the induced coordinates on T M will be denoted by (x, y) = (x^{i}, y^{i}). Let {0} be the zero-section of the tangent bundle and X (T M) the Lie algebra of vector fields on T M. A main structure of T M is the tangent structure J = \frac{∂}{∂y^{i}} ⊗dx^{i}.

Fix also a symmetric linear connection ∇ on M and let S_{∇} ∈ X (T M) its semispray; if ∇ has the local coefficients Γ_{jk}^{i} (x), then S_{∇} has the local expression S_{∇} = y^{i} \frac{∂}{∂x^{i}} − 2G^{i}(x, y)\frac{∂}{∂y^{i}} with

(2.1)  G^{i} = \frac{1}{2}Γ_{jk}^{i} y^{j} y^{k}.

We have

(2.2)  J(S_{∇} ) = ℂ

with ℂ = y^{i} \frac{∂}{∂y^{i}} the Liouville vector field of T M. This vector field is useful in characterizations of homogeneous objects on T M ∖ {0}, [2, p. 163]; namely let k be an integer, then a vector field X is k-homogeneous if L_{ℂ}X = (k − 1)X, a p-form ω is k-homogeneous if L_{ℂ}ω = kω and a (1, 1)-tensor field L is k-homogeneous if L_{ℂ}L = (k − 1)L. Here L_{Z} is the Lie derivative with respect to the vector field Z; the vector fields \{ \frac{∂}{∂y^{i}} \} are 0-homogeneous.

In fact S_{∇} is more than a semispray; it is a spray due to the 2-homogeneity of the coefficients G^{i} of (2.1). Then S_{∇} has the reduced form S_{∇} = y^{i} \frac{δ}{δx^{i}} where

\frac{δ}{δx^{i}} = \frac{∂}{∂x^{i}} − N_{i}^{j} \frac{∂}{∂y^{j}}

with N_{j}^{i} = \frac{∂G^{i}}{∂y^{j}} = Γ_{jk}^{i} y^{k}. We derive a new local basis ( \frac{δ}{δx^{i}} , \frac{∂}{∂y^{i}} ) in X (T M), called Berwald, which is adapted to our computations and has the dual basis (dx^{i}, δy^{i} = dy^{i} + N_{j}^{i}dx^{j} ). This decomposition of the iterated tangent bundle T (T M) produces [2, p. 165]:

i1) the horizontal projector h = \frac{δ}{δx^{i}} ⊗ dx^{i},

ii1) the vertical projector v = \frac{∂}{∂y^{i}} ⊗ δy^{i}, and the following distributions and codistributions:

i2) the horizontal distribution H(T M) = span\{ \frac{δ}{δx^{i}} \} = h(X (T M)) and its annihilator H^{∗}(T M) = span{δy^{i}},

ii2) the vertical distribution V (T M) = span\{ \frac{∂}{∂y^{i}} \} = v(X (T M)) and its annihilator V^{∗}(T M) = span{dx^{i}}.

Therefore S_{∇} is a horizontal vector field while ℂ is a vertical vector field. Let ∇_{S} be the dynamical covariant derivative induced on T M by S_{∇} , [2, p. 167].

Inspired by the defining property of the Levi-Civita connection, we introduce:

Definition 2.1. The symmetric linear connection ∇ is said to be tangent metrizable if there exists a Riemannian metric g on T M, of Sasaki type

g = g_{ij} dx^{i} ⊗ dx^{j} + g_{ij} δy^{i} ⊗ δy^{j}

such that ∇_{S}g = 0 which means the tangent Christoffel formula

(2.3)  S(g_{ij} ) = N_{i}^{k}g_{kj} + N_{j}^{k}g_{ki}.

Our approach in finding characterizations of tangent metrizability follows closely the techniques of Bucataru and Dahl from [2], where, among others, the following type of metrizability is studied:

Definition 2.2. [2, p. 178] The spray S is Finsler metrizable if there exists a 2-homogeneous Lagrangian L : T M → ℝ such that L_{S}(d_{J}L) = dL. The spray is called projective metrizable if the above Lagrangian is 1-homogeneous.

Here d_{J} is the exterior derivative induced by J on forms of T M; then d_{J}L(X) = dL(JX) = (JX)(L) for any X ∈ X (T M). The fundamentals of Finsler geometry are excellently exposed in [1].

The characterization of Finsler metrizability is obtained in [2] in terms of semibasic 1-forms i.e., 1-forms on T M with local expression θ = θ_{i}(x, y)dx^{i}:

Theorem 2.1. [2, p. 177] The spray S is Finsler metrizable if and only if there exists a 1-homogeneous semi-basic 1-form θ on the slit tangent bundle T M ∖ {0} such that the following Helmoltz conditions hold

(1) θ is d_{h}-closed: d_{h}θ = 0,

(2) θ is d_{J}-closed: d_{J}θ = 0,

(3) dθ is covariant constant with respect to ∇_{S}: ∇_{S}(dθ) = 0.

Then θ is called the Poincaré–Cartan 1-form of the Lagrangian L.

Let us point out that our condition (2.3) corresponds to the relation (3) above. Therefore we apply Theorem 2.1 and to this aim let ω = dθ. Recall also:

Definition 2.3. [2, p. 164] The semibasic 1-form θ is called nondegenerate if ω is a symplectic form on T M ∖ {0}.

Recall that a pair (ω, K) with a symplectic form and a compatible almost complex structure yields a Riemannian metric g through the formula [4, p. 86]:

(2.4)  g(·, ·) = ω(·, K·).

The required compatibility is [4, p. 90]:

(2.5)  ω(KX, KY) = ω(X, Y).

But ∇ (equivalently S_{∇} ) yields an almost complex structure 𝔽, [2, p. 166]:

(2.6)  𝔽( \frac{δ}{δx^{i}} ) = −\frac{∂}{∂y^{i}} , 𝔽( \frac{∂}{∂y^{i}} ) = \frac{δ}{δx^{i}}

or, in the Berwald basis

𝔽 = \frac{δ}{δx^{i}} ⊗ δy^{i} − \frac{∂}{∂y^{i}} ⊗ dx^{i}.

Now, we are able to state the first main result of this note.

Proposition 2.1. Let ∇ be a symmetric linear connection on M such that the associated spray S_{∇} is Finsler metrizable via the semibasic 1-form θ. If θ is nondegenerate i.e., the determinant det( \frac{∂θ_{i}}{∂y^{j}} ) is nonvanishing, then ∇ is tangent metrizable.

Proof. We need to prove two facts.

(I) ω and 𝔽 are compatible, but this is a consequence of relations (1) and (2) from Theorem 2.1. Indeed, after [2, p. 172], we have ω = a_{ij} dx^{j} ∧dx^{i} +2g_{ij} δy^{j} ∧dx^{i} and d_{h}θ = a_{ij} dx^{j} ∧ dx^{i}, d_{J}θ = (g_{ij} − g_{ji} )dx^{j} ∧ dx^{i} with

a_{ij} = \frac{1}{2}( \frac{δθ_{i}}{δx^{j}} − \frac{δθ_{j}}{δx^{i}} ) , g_{ij} = \frac{1}{2} \frac{∂θ_{i}}{∂y^{j}} .

A direct computation gives

(i)  ω( 𝔽\frac{δ}{δx^{i}} , 𝔽\frac{δ}{δx^{j}} ) = 0 = a_{ij} = ω( \frac{δ}{δx^{i}} , \frac{δ}{δx^{j}} ) ,
(ii)  ω( 𝔽\frac{∂}{∂y^{i}} , 𝔽\frac{∂}{∂y^{j}} ) = a_{ij} = 0 = ω( \frac{∂}{∂y^{i}} , \frac{∂}{∂y^{j}} ) ,
(iii)  ω( 𝔽\frac{δ}{δx^{i}} , 𝔽\frac{∂}{∂y^{j}} ) = −g_{ji} = −g_{ij} = ω( \frac{δ}{δx^{i}} , \frac{∂}{∂y^{j}} ) .
(II) ∇_{S}g = 0. The identity (3) reads ∇_{S}ω = 0 and we have ∇_{S}𝔽 = 0 from

[2, p. 169]; these two relations yield ∇_{S}g = 0 because of (2.4).

Let us remark that (i) and (ii) mean that the distributions H(T M) and V (T M) are Lagrangian for ω.

The matrix of ω in the dual Berwald basis (dx^{i}, δy^{j} ) is

ω = ( \begin{matrix}0 & −g_{ij} \\ g_{ij} & 0 \end{matrix} )

and then ω is non-degenerate if and only if det(g) ≠ 0. The matrix of 𝔽 is

𝔽 = ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} )

and then the corresponding metric g is of Sasaki type

g = ( \begin{matrix}0 & −g_{ij} \\ g_{ij} & 0 \end{matrix} ) · ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ) = ( \begin{matrix}g_{ij} & 0 \\ 0 & g_{ij} \end{matrix} ) .  □

In this framework, lifting to the tangent bundle the geodesic equations of ∇ i.e., the flow equations of S_{∇} we arrive at exactly the Euler–Lagrange equations of the Lagrangian L_{∇,θ} = g(S, S) = g(ℂ, ℂ) = ω(ℂ, 𝔽(ℂ)) which can be thought of as the energy of g. A dual relation to (2.2) is

(2.7)  𝔽(ℂ) = S_{∇}

and then

L_{∇,θ} = ω(ℂ, S_{∇} ) = dθ(ℂ, S_{∇} ) = \frac{1}{2}\{ℂ(θ(S_{∇} )) − S_{∇} (θ(ℂ)) − θ([ℂ, S_{∇} ])\}.

Since θ is semibasic, we have θ(ℂ) = 0 while the 2-homogeneity of S_{∇} reads

[ℂ, S_{∇} ] = S_{∇}

and then

L_{∇,θ} = ℂ( \frac{1}{2}θ(S_{∇} )) − \frac{1}{2}θ(S_{∇} )

which recover the classical formula for the Hamiltonian H associated to a Lagrangian L

H = y^{i} \frac{∂L}{∂y^{i}} − L = ℂ(L) − L.

But again the 2-homogeneity of S_{∇} : ℂ(θ(S_{∇} )) = 2θ(S_{∇} ) gives the final expression L_{∇,θ} = \frac{1}{2}θ(S_{∇} ) = \frac{1}{2}θ_{i}(x, y)y^{i} which is 2-homogeneous since θ is 1-homogeneous. A similar formula can be derived by using the 𝔽-adjoint of θ

θ^{∗} = 𝔽^{∗}(θ) = \frac{1}{2}θ_{i}δy^{i}

(𝔽^{∗} is the version of 𝔽 on 1-forms), namely L_{∇,θ} = θ^{∗}(ℂ). Let us point out that the Euler characterization of 1-homogeneity for θ

θ_{i} = \frac{∂θ_{i}}{∂y^{j}} y^{j}

means that θ_{i} = 2g_{ij} y^{j}, and then L has the well-known expression L = g_{ij} y^{i}y^{j} .

The final answer to the metrizability problem. Now, we are ready for a return to the initial problem namely the metrizability of ∇. If we assume the smoothness of our tools on the whole T M, then the 1-homogeneity of θ means that the components θ_{i} are homogeneous polynomials of degree 1 in y and then g is in fact a Riemannian metric on the base manifold M i.e., g = g(x). In conclusion, our proposed approach for the metrizability problem of a linear connection ∇ consists in two steps:

(1) we lift from M to the tangent bundle T M and study the Finsler metrizability of S_{∇} obtaining the metric g,

(2) we return to the basis manifold M under the homogeneity hypothesis, obtaining the desired metric g = g(x).

Examples 2.1. (1) Let on M = ℝ^{2} the symmetric linear connection with the only nonzero components, [19, p. 513]

Γ_{11}^{1} = \frac{x^{1}}{(x^{1})^{2} + 1} , Γ_{22}^{2} = \frac{x^{2}}{(x^{2})^{2} + 1} .

In the cited paper, it is proved that ∇ is the Levi-Civita connection for a metric g on the base manifold and this g is computed. In the present framework we derive that ∇ is tangent metrizable with the semibasic 1-form

θ^{1} = 2b_{2}[(x^{1})^{2} + 1]y^{1} + b_{1} \sqrt{[(x^{1})^{2} + 1][(x^{2})^{2} + 1]} y^{2},
θ^{2} = b_{1} \sqrt{[(x^{1})^{2} + 1][(x^{2})^{2} + 1]} y^{1} + 2b_{3}[(x^{2})^{2} + 1]y^{2}

with b_{i} real parameters.

(2) Examples of non-Finsler metrizable sprays are in [3].

It is natural to ask about ω^{∗} = dθ^{∗}. A straightforward computation yields

ω^{∗} = \frac{1}{2}R_{ij}^{k} θ_{k}dx^{i} ∧ dx^{j} + \frac{1}{2}θ_{i|j} dx^{i} ∧ δy^{j}

where (R_{ij}^{a} ) are the components of the curvature of nonlinear connection H(T M), [2, p. 166]

R_{ij}^{a} = \frac{δN_{i}^{a}}{δx^{j}} − \frac{δN_{j}^{a}}{δx^{i}} and θ_{j|i} = \frac{δθ_{j}}{δx^{i}} − \frac{∂N_{i}^{k}}{∂y^{j}} θ_{k}.

In Finsler (particularly Riemannian) geometry, by using the Cartan (particularly Levi-Civita) linear connection, the coefficient θ_{i|j} vanish and then dθ is the unique associated symplectic form. A Finslerian spray S with vanishing curvature R is called R-flat [7]. So, for an R-flat Finsler metric (particularly flat Riemannian metric) we have that θ^{∗} is closed.

Let us end this section by making several relationships with some previous works.

(a) Usually, one seeks the solution of the metrizability problem bearing in mind the holonomy groups (or algebra) according to the first fundamental paper on this subject namely [17]. So, in [19, p. 514] are studied symmetric bilinear forms G with

(2.8)  G(AX, Y) + G(X, AY) = 0

for all A in the holonomy group of a given point of M. Since 𝔽 of (2.6) is an almost complex structure, the compatibility relation (2.5) reads:

(2.9)  g(𝔽X, Y) + g(X, 𝔽Y) = 0

and we remark that the formulae (2.8) and (2.9) are formally the same. The essential difference between these equations is that formula (2.9) is in a close relation to the curvature (and there is no such condition if the curvature is zero) while (2.8) is just expressing some symmetry of the metric (and survives even in the case when R = 0). In fact, formula (2.8) corresponds rather to one of the Helmholtz condition and its generalization involving the Jacobi endomorphism and its derivatives.

(b) In [7, p. 100] it is pointed out that 𝔽 of (2.6) is not homogeneous and a new almost complex structure is considered

𝔽_{S}( \frac{δ}{δx^{i}} ) = −F \frac{∂}{∂y^{i}} , 𝔽_{S}( \frac{∂}{∂y^{i}} ) = \frac{1}{F} \frac{δ}{δx^{i}}

where F is the Finslerian fundamental function corresponding to the Lagrangian L_{∇,θ} constructed above via L_{∇,θ} = F^{2}. This new almost complex structure is 0homogeneous [7, p. 100], and a computation similar to that given in the proof of Proposition 2.1 yields that ω and 𝔽_{S} are still compatible. The formula corresponding to (2.7) is 𝔽_{S}(ℂ) = \frac{1}{F} S_{∇} and then the energy of the new metric g_{S} = ω(·, 𝔽_{S}·) is

L_{∇,θ,S} = \frac{1}{F} L_{∇,θ} = F = \sqrt{θ_{i}(x, y)y^{i}} = \sqrt{g_{ij} y^{i}y^{j}}.

This 1-homogeneous Lagrangian corresponds to the projective metrizability of S_{∇} .

(c) If ω and 𝔽 are not compatible, then, similar to relation (11) of [13, p. 8395], we replace ω with ω_{s}(·, ·) = \frac{1}{2}\{ω(𝔽·, 𝔽·) + ω(·, ·)\}.

(d) Also, from relation (29) from [13, p. 8396], we get that S_{∇} is, at the same time:

– a Killing vector field for g,

– an infinitesimal symplectomorphism for ω,

– an infinitesimal automorphism for 𝔽.

According to [4, p. 22] for an exact symplectic form ω = dθ, there exists a unique vector field ξ such that i_{ξ}ω = θ. For our case we have ξ = ℂ so that we can say that all semisprays have an universal symplectic potential vector field namelyℂ.

(e) Again from [7, p. 100], the almost complex structure 𝔽 is integrable if and only if the spray S_{∇} is R-flat which means that ∇ is with vanishing curvature. In this case, the pair (g, 𝔽) is a Kähler structure on T M ∖ {0} for which ω = dθ is the Kähler (or fundamental) form.

3. The relationship with the Courant structure of big TM. Associated Dirac structures

The interplay between vector fields and 1-forms of the previous section makes necessary the consideration of the manifold T^{big}M := T M ⊕ T^{∗}M. This manifold is the total space of a natural vector bundle π : T^{big}M → M; so T^{big}M is called the big tangent bundle of M [18], and is endowed with the Courant structure (⟨ , ⟩, [ ,]) [6]:

1. the (neutral) inner product ⟨(X, α), (Y, β)⟩ = \frac{1}{2}(β(X) + α(Y))

2. the (skew-symmetric) Courant bracket

[(X, α), (Y, β)] = ( [X, Y], L_{X} β − L_{Y}α − \frac{1}{2} d(β(X) − α(Y))) .

The same manifold T M ⊕ T^{∗}M is called sometimes the Pontryagin bundle of M [11].

Going now with the similar structure by considering T M instead of M, we derive some ways to express the Lagrangian associated to a metrizable spray:

Proposition 3.1. Let S be a spray which is Finsler metrizable through the semibasic 1-form θ. Then the corresponding 2-homogeneous Lagrangian is

L = ⟨(ℂ, θ), (S, θ)⟩ = ⟨(ℂ, θ^{∗}), (S, θ^{∗})⟩ = \frac{1}{2}⟨(ℂ, θ), (S, θ^{∗})⟩.

Proof. The above formulae are direct consequences of the following expressions of L

⟨( \frac{∂}{∂y^{i}} , θ) , (S, θ)⟩ = ⟨( \frac{δ}{δx^{i}} , θ) , (S, θ^{∗})⟩
= ⟨( \frac{∂}{∂y^{i}} , θ^{∗}) , (ℂ, θ)⟩ = ⟨( \frac{δ}{δx^{i}} , θ^{∗}) , (ℂ, θ^{∗})⟩.  □

Another interesting relation is

⟨( \frac{∂}{∂y^{i}} , θ^{∗}) , ( \frac{δ}{δx^{j}} , θ) ⟩ = 0,

which means that the distributions (V (T M), θ^{∗}), (H(T M), θ) are ⟨ , ⟩-orthogonal on T^{big}T M.

The Courant bracket on T^{big}T M gives the following version of Theorem 5.4 from [2, p. 177]:

Proposition 3.2. Let S be a spray on M. Then S is a Lagrangian vector field, induced by a k-homogeneous Lagrangian, if and only if there exists a (k − 1)- homogeneous semibasic 1-form θ ∈ Ω^{1}(T M ∖ {0}) such that

[(ℂ, θ), (S, θ)] = ( S, (k − 1)θ + ( \frac{k}{2} − 1) L_{S}θ) .

It results in particular that S is:

– Finsler metrizable if and only if there exists a 1-homogeneous semibasic 1form θ such that [(ℂ, θ), (S, θ)] = (S, θ),

– projective metrizable if and only if there exists a 0-homogeneous semibasic 1-form θ such that [(ℂ, θ), (S, θ)] = (S, −\frac{1}{2}L_{S}θ).

Recall [18], that [ ,]-integrable, ⟨ , ⟩-isotropic subbundles of T^{big}M of maximal rank n are called Dirac structures; so, our next aim is to derive Dirac structures in the present framework.

With a computation similar to that of the first section we get

L_{\frac{δ}{δx^{i}}} θ^{∗} = −R_{ij}^{a} θ_{a}dx^{j} + θ_{i|j} δy^{j}

and then

[( \frac{δ}{δx^{i}} , θ^{∗}) , ( \frac{δ}{δx^{j}} , θ^{∗}) ] = ( R_{ij}^{k} \frac{∂}{∂y^{k}} , (R_{ja}^{b} − R_{ia}^{b} )θ_{b}dx^{a} + (θ_{i|u} − θ_{j|u} )δy^{u}) .

For a Finslerian (particularly Riemannian) spray S the above relation becomes

[( \frac{δ}{δx^{i}} , θ^{∗}) , ( \frac{δ}{δx^{j}} , θ^{∗}) ] = ( R_{ij}^{k} \frac{∂}{∂y^{k}} , (R_{ja}^{b} − R_{ia}^{b} )θ_{b}dx^{a}) .

So, for a R-flat Finslerian metric (particularly flat Riemannian metric) the subbundle (H(T M), θ^{∗}) can be considered as a \frac{1}{2}-Dirac structure on T^{big}T M.

From

L_{\frac{δ}{δx^{i}}} dx^{j} = L_{\frac{∂}{∂y^{i}}} dx^{j} = 0, L_{\frac{δ}{δx^{i}}} δy^{j} = −R_{ia}^{j} dx^{a} − \frac{∂N_{i}^{j}}{∂y^{a}} δy^{a}, L_{\frac{∂}{∂y^{i}}} δy^{j} = \frac{∂N_{i}^{j}}{∂y^{a}} δy^{a}

we compute

[( \frac{δ}{δx^{i}} , dx^{j} ) , ( \frac{δ}{δx^{k}} , dx^{l}) ] = ( R_{ik}^{a} \frac{∂}{∂y^{a}} , 0) , [( \frac{∂}{∂y^{i}} , δy^{j} ) , ( \frac{∂}{∂y^{k}} , δy^{l}) ]

= ( 0, ( \frac{∂N_{i}^{l}}{∂y^{a}} − \frac{∂N_{k}^{j}}{∂y^{a}} ) δy^{a})

so (H(T M), V^{∗}(T M)) is Courant closed if and only if the Finslerian spray S (particularly Riemannian metric g) is R-flat (flat) and (V (T M), H^{∗}(T M)) is always Courant closed.

Another main result of this section is:

Theorem 3.1. V_{D}(S) = (V (T M), V^{∗}(T M)) is a Dirac structure on T^{big}T M. H_{D}(S) = (H(T M), H^{∗}(T M)) is a Dirac structure on T^{big}T M if and only if the Finslerian spray S, in particular the Riemannian metric g, is R-flat, in particular is flat.

Proof. We have

⟨( \frac{∂}{∂y^{i}} , dx^{j} ) , ( \frac{∂}{∂y^{k}} , dx^{l}) ⟩ = 0, ⟨( \frac{δ}{δx^{i}} , δy^{j} ) , ( \frac{δ}{δx^{k}} , δy^{l}) ⟩ = 0

respectively

[( \frac{∂}{∂y^{i}} , dx^{j} ) , ( \frac{∂}{∂y^{k}} , dx^{l}) ] = (0, 0),

[( \frac{δ}{δx^{i}} , δy^{j} ) , ( \frac{δ}{δx^{k}} , δy^{l}) ] = ( R_{ik}^{a} \frac{∂}{∂y^{a}} , (R_{ka}^{j} − R_{ia}^{l} )dx^{a} + ( \frac{∂N_{k}^{j}}{∂y^{a}} − \frac{∂N_{i}^{l}}{∂y^{a}} ) δy^{a})

which yields the conclusions. □

It results that an R-flat Finslerian spray or a flat Riemannian metric yields a decomposition of T^{big}T M in complementary Dirac structures:

(3.1)  T^{big}T M = V_{D}(S) ⊕ H_{D}(S)

in analogy with the decomposition of the iterated tangent bundle

T (T M) = V (T M) ⊕ H(T M).

From

⟨( \frac{∂}{∂y^{k}} , dx^{l}) , ( \frac{δ}{δx^{i}} , δy^{j} ) ⟩ = \frac{1}{2}(δ_{i}^{l} + δ_{k}^{j})

the decomposition (3.1) is not ⟨, ⟩-orthogonal. Recall, after [10] that a pair of complementary Dirac subspaces is called a reflector. So, the decomposition (3.1) provides T^{big}T M with a reflector.

Recall also that if L is a Dirac structure on M, then the triple (L, [ ,]|_{L}, pr : L → T M) is a Lie algebroid on M [6, p. 645]. Therefore, we get two Lie algebroids over T M provided by a flat Riemannian metric, or more generally R-flat Finsler spray.

4. The relationship with the almost symplectic structure of big T M

In the first section we have a symplectic structure on T M associated with our framework; let us remark that the big tangent bundle has a nondegenerate skew-symmetric 2-form [18] Ω((X, α), (Y, β)) = \frac{1}{2}(β(X) − α(Y)). Our aim in this section is to use this almost symplectic structure of big tangent bundle for our computations.

From

Ω( ( \frac{∂}{∂y^{i}} , θ) , ( \frac{∂}{∂y^{j}} , θ) ) = 0, Ω( ( \frac{δ}{δx^{i}} , θ^{∗}) , ( \frac{δ}{δx^{j}} , θ^{∗}) ) = 0

it results that (V (T M), θ) and (H(T M), θ^{∗}) are Ω-isotropic subbundles of T^{big}T M, while from

Ω( ( \frac{∂}{∂y^{i}} , dx^{j} ) , ( \frac{∂}{∂y^{k}} , dx^{l}) ) = 0, Ω( ( \frac{δ}{δx^{i}} , δy^{j} ) , ( \frac{δ}{δx^{k}} , δy^{l}) ) = 0

we get that the complementary Dirac structures V_{D}(S) and H_{D}(S) are Ω-Lagran-gian subbundles of T^{big}T M.

References

1. D. Bao, S.-S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, Grad. Texts Math. 200, Springer-Verlag, New York, 2000, MR1747675 (2001g:53130)

2. I. Bucataru, M. F. Dahl, Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations, J. Geom. Mech. 1(2) (2009), 159–180, MR2525756 (2010h:58025)

3. I. Bucataru, Z. Muzsnay, Projective metrizability and formal integrability, SIGMA 7 (2011), 114, 22 pages, http://www.emis.de/journals/SIGMA/2011/114/sigma11-114.pdf, MR2861227

4. A. Cannas da Silva, Lectures on Symplectic Geometry, Lect. Notes Math. 1764, Springer-Verlag, Berlin, 2001, MR1853077 (2002i:53105)

5. M. Cocos, A note on symmetric connections, J. Geom. Phys. 56(3) (2006), 337–343, MR2171888 (2006f:53029)

6. T. J. Courant, Dirac manifolds, Trans. Am. Math. Soc. 319(2) (1990), 631–661, MR0998124 (90m:58065)

7. M. Crampin, Kähler and para-Kähler structures associated with Finsler spaces of non-zero constant flag curvature, Houston J. Math. 34(1) (2008), 99–114. MR2383698 (2009c:53104)

8. M. Crasmareanu, Dirac structures from Lie integrability, Int. J. Geom. Methods Mod. Phys. 9(4) (2012), 1220005, 7 pp., MR2917304

9. L. P. Eisenhart, O. Veblen, The Riemann Geometry and its Generalization, Proc. Natl. Acad. Sci. USA 8(2) (1922), 19–23.

10. G. R. Jensen, M. Rigoli, Neutral surfaces in neutral four-spaces, Matematiche (Catania) 45(2) (1990), 407–443, MR1152942 (93d:53084)

11. M. Jotz, T. S. Ratiu, J. Śniatycki, Singular reduction of Dirac structures, Trans. Am. Math. Soc. 363 (2011), 2967–3013, MR2775795 (2012d:53254)

12. D. A. Lizárraga, J. M. Sosa, Control of mechanical systems on Lie groups based on vertically transverse functions, Math. Control Signals Systems 20(2) (2008), 111–133, MR2411415 (2009c:93038)

13. G. Marmo, G. Morandi, A. Simoni, F. Ventriglia, Alternative structures and bi-Hamiltonian systems, J. Phys. A 35(40) (2002), 8393–8406, MR1947538 (2003m:37105)

14. T. Mestdag, W. Sarlet, M. Crampin, The inverse problem for Lagrangian systems with certain non-conservative forces, Differ. Geom. Appl. 29(1) (2011), 55–72, MR2784288 (2012g:70027)

15. D. C. Robinson, The real geometry of holomorphic four-metrics, J. Math. Phys. 43(4) (2002), 2015–2028, MR1892765 (2003a:53106)

16. W. Sarlet, G. Prince, T. Mestdag, O. Krupková, Time-dependent kinetic energy metrics for Lagrangians of electromagnetic type, J. Phys. A 45(8) (2012), 085208, 13 pp., MR2897016

17. B. G. Schmidt, Conditions on a connection to be a metric connection, Commun. Math. Phys. 29 (1973), 55–59, MR0322726 (48#1087)

18. I. Vaisman, Isotropic subbundles of T M ⊕ T^{∗}M, Int. J. Geom. Methods Mod. Phys. 4(3) (2007), 487–516, MR2343378 (2008g:53102)

19. A. Vanžurová, Metrization problem for linear connections and holonomy algebras, Arch. Math. (Brno), 44(5) (2008), 511–521, MR2501581 (2010b:53079)

20. H. Yoshimura, J. E. Marsden, Dirac structures in Lagrangian mechanics. I. Implicit Lagrangian systems, J. Geom. Phys. 57(1) (2006), 133–156, MR2265464 (2007j:37087)

Faculty of Mathematics, University “Al. I. Cuza" (Received 10 11 2012) Iaşi, Romania

 mcrasm@uaic.ro