Transitive Courant algebroids

We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal Whitney sum $E\oplus C$ where E is a given Courant algebroid and C is a flat, pseudo- Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive Courant algebroid, i.e., a Courant algebroid with a surjective anchor, and describe a class of transitive Courant algebroids which are Whitney sums of a Courant subalgebroid with neutral metric and Courant-like bracket and a pseudo-Euclidean vector bundle with a flat, metric connection. In particular, this class contains all the transitive Courant algebroids of minimal rank; for these, the flat term mentioned above is zero. The results extend to regular Courant algebroids, i.e., Courant algebroids with a constant rank anchor. The paper ends with miscellaneous remarks and an appendix on Dirac linear spaces.


The basics of Courant algebroids
The framework of this note is the C ∞ -category. In the literature, there are two notions of a Courant algebroid, which include a skew-symmetric and a non-skew-symmetric bracket, respectively. These notions are the result of an effort to unify the Courant bracket and the Manin bracket [4].
We start with the definition of a non-skew-symmetric Courant algebroid [6,7], with the simplifications indicated in [8].
Let us also consider the skew-symmetric part of the product: Proof. Property a) follows from the comparison of the results of expressing (ρe)(g(f e 1 , e 2 )) in two ways, first by applying property 1) of Definition 1.1 straightforwardly, second by using the Leibniz rule for the vector field ρe applied to the product f g(e 1 , e 2 ) and then property 1) for g(e 1 , e 2 ). Property b) follows from a) by using (1.2) and the fact that, on functions, ∂ satisfies the Leibniz rule.
Notice that properties a) and b) show that a Courant algebroid product is an operator of the local type (i.e., (e 1 ⋆e 2 )(x) depends only on the restrictions of e 1 , e 2 to a neighborhood of the point x ∈ M). Now, let us denote Since any function f ∈ C ∞ (M) may locally be written as where e 1 is not g-isotropic, equation (1.9) and property 3) of Definition 1.1 imply the first part of c). The second part of c) follows from the first part and the relation (1.2). In order to get property d) (where the right hand side is a Lie bracket of vector fields on M) we start with (1.3), written as (ρe 2 )f = 2g(∂f, e 2 ), and apply ρe 1 , while using 1) of Definition 1.1 and (1.3) again. The result is (1.10) (ρe 1 )(ρe 2 )f = 2g(e 1 ⋆ (∂f ), e 2 ) + ρ(e 1 ⋆ e 2 )f.
Since f is an arbitrary function, we got precisely d).
Finally, from b) we get and, if we use d) in the two sides of the previous relation while assuming g(e 1 , e 2 ) = 0, we deduce the property e). Q.e.d.
Remark 1.1 Property c) implies that the skew-symmetric part of a Courant algebroid product satisfies the property Property e) is equivalent with g(∂f, ∂g) = 0, ∀f, g ∈ C ∞ (M) i.e., with the fact that im ∂ x is a g-isotropic subspace of the fiber E x , ∀x ∈ M. We also note that property e) is implied by b) and d). Finally, the computation used in the proof of d) is reversible in the sense that d) and formula (1.10) imply property c) as well as its consequence (1.11).
Remark 1.2 If ρ = 0 we have ∂ = 0 and the Courant algebroid is just a bundle of Lie algebras with a pseudo-Euclidean metric g that is invariant for the Lie algebra structure of each fiber.
From Definition 1.1 and Proposition 1.1 it follows that the skew-symmetric part (1.6) of a product ⋆ has the properties indicated by the following proposition.
For simplicity, in what follows we will use the language provided by the next definitions is a pseudo-Euclidean vector bundle over the manifold M, which is such that im ρ is a (generalized) foliation and the corresponding morphism ∂ : T * M → E defined by (1.1) has a g-isotropic image. A triple (E, g, ρ) where ρ is a Courant anchor will be called a Courant vector bundle. A Courant vector bundle endowed with a skew-symmetric bracket [ , ] on ΓE, which satisfies properties v) and i) (therefore, also, iv) and ii) of a skewsymmetric Courant algebroid will be called a pre-Courant algebroid.
Notice that, since, ∀x ∈ M, both im ρ and im ∂ have the dimension equal to rank ρ, a Courant vector bundle must satisfy the condition rank ρ ≤ b ≤ (1/2)rank E, where b is the smallest between the positive-negative inertia indices of g. Furthermore, in view of Remark 1.1, formula (1.11) holds for any pre-Courant algebroid. Using (1.11) it is easy to check that any pre-Courant algebroid satisfies property iii) of Proposition 1.2 if at least one of the arguments e a ∈ im ∂ (a = 1, 2, 3) (the cross sections of the subbundle im ∂ are locally spanned over C ∞ (M) by cross sections of the form ∂f , (f ∈ C ∞ (M))).
By Proposition 1.2, the skew-symmetric part of a Courant algebroid product yields a skew-symmetric Courant algebroid bracket. The converse is also true: Proposition 1.3 (E.g., [6]). If the bracket [ , ] satisfies the properties of a skew-symmetric Courant algebroid, the product ⋆ defined by (1.7) satisfies the properties of a Courant algebroid. Proof. The difference of two brackets of pre-Courant algebroid structures is a form λ ∈ Γ(∧ 2 E * ⊗ E) because of property iv), Proposition 1.2. The indicated conditions for λ are equivalent with properties i) and v) of the same Proposition, respectively. Notice that ρ • λ = 0 is equivalent with therefore, in view of the skew-symmetry of Λ, also equivalent with Finally, a technical computation shows that property iii) is equivalent with (1.17), (1.18). Q.e.d.

Courant brackets and metric connections
We will get further insight into the structure of the bracket of a Courant algebroid by using a metric connection ∇ on the pseudo-Euclidean bundle (E, g), which means that (2.1) X(g(e 1 , e 2 )) = g(∇ X e 1 , e 2 ) + g(e 1 , ∇ X e 2 ) (X ∈ ΓT M).
If we also have the morphism ρ : E → T M, we define the ρ-torsion T (∇,ρ) ∈ Γ(∧ 2 E * ⊗ T M) by the formula Now, we can prove where γ is defined by the equality defines a skew-symmetric bracket on ΓE that satisfies property v) of Proposition 1.2.
3) The bracket (2.5) also satisfies property i) of Proposition 1.2 iff the following two equalities hold where T (∇,ρ) is the ρ-torsion of ∇. In particular, if (E, g, ρ) has a metric connection ∇ with zero ρ-torsion, the bracket (2.3) satisfies property i).
Proof. First, let us notice that formula (2.4) actually defines γ because if this formula holds for e ∈ ΓE it also holds for f e, ∀f ∈ C ∞ (M), and since g is non degenerate. Then, a technical calculation, which takes into account the metric property (2.1) shows that the bracket (2.3) satisfies property v). Now, the difference β between two brackets that satisfy v) is such that the corresponding B given by (2.6) is totally skew-symmetric, and β ∈ Γ(∧ 2 E * ⊗ E), respectively B ∈ ∧ 3 E * , because v) implies property iv) of Proposition 1.2. These remarks justify the general formula (2.5).
where J is defined by (1.12), and denote by an index 0 the same expression for the bracket [ , ] 0 . Assuming that property i) holds, it follows that where R ∇ is the curvature of the connection ∇ and ∇γ is defined as if γ would be a tensor. These formulas express the condition for the bracket (2.5) to satisfy property iii), C = 0, by means of the metric connection ∇. In particular, if there exists a metric connection of zero ρ-torsion, the corresponding bracket (2.3), which satisfies i), also satisfies iii) iff C 0 = 0, and this condition reminds us the Bianchi identity for a linear connection with torsion on a differentiable manifold M.
As an application of the results given in this section we have ) be a pre-Courant algebroid. Then, for any pseudo-Euclidean vector bundle (C, g 0 ) over M and any metric connection ∇ on C, the brackets where e, e 1 , e 2 ∈ ΓE, c, c 1 , c 2 ∈ ΓC and γ 0 (c 1 , c 2 ) ∈ ΓE is defined by define a pre-Courant algebroid structure on (E ⊕ C, g ⊕ g 0 , ρ ⊕ 0). If the original algebroid is a Courant algebroid, (2.14) yields a Courant algebroid structure iff the connection ∇ is flat.
Proof. Straightforward checks show that the brackets (2.14) satisfy property v) of a pre-Courant algebroid. For a full justification of property i) we must also notice that ρ(γ 0 (c 1 , c 2 ) = 0. This follows since (2.15) implies For the last assertion of the proposition we refer to property iii) of a Courant algebroid. For arguments e 1 , e 2 , e 3 , iii) holds if E is a Courant algebroid and for arguments c 1 , c 2 , c 3 iii) follows from ρ(γ 0 (c 1 , c 2 )) = 0. For arguments e 1 , e 2 , c, iii) is just R ∇ (ρe 1 , ρe 2 )c = 0, where R ∇ is the curvature of ∇. Finally, for arguments e, c 1 , c 2 , iii) means Since for a pre-Courant algebroid property iv) holds, the left hand side of the previous equality is C ∞ (M)-linear and it suffices to check it for a local basis of C. If connection ∇ is flat, C has local ∇-parallel bases and (2.16) obviously holds. Q.e.d.

Transitive and regular Courant algebroids
In this section we determine the structure of the transitive Courant algebroids, i.e., Courant algebroids with a surjective anchor. The results may then be extended to regular Courant algebroids, i.e., Courant algebroids with a constant rank anchor. When this paper was ready, I was informed that the transitive Courant algebroids were also determined by P.Ševera in unpublished correspondence with A. Weinstein, without metric connections (see Remark 3.2 later on).
Let (E, g, ρ) be a Courant vector bundle with a surjective anchor ρ. Then, K = ker ρ is a (regular) subbundle of E of rank k = r − n, where r = rank E and n = dim M, and, if K ⊥g is the g-orthogonal subbundle of K, dim(K ∩K ⊥g ) = r −s, where s = dim(K +K ⊥g ). In view of the properties of a Courant anchor dim(im ∂) = n and im ∂ ⊆ (K ∩ K ⊥g ) whence, it follows easily that s = k. Accordingly, K is a g-coisotropic subbundle of E and K ⊥g = im ∂ ⊆ K.
In this situation, it is known that there exists an isotropic, complementary subbundle Q of K in E and a complementary subbundle C of im ∂ in K such that is a g-orthogonal decomposition, the restriction of g to C is non-degenerate, and the restriction of g to P is non-degenerate and neutral (i.e., of signature zero). We will say that P is a neutral completion of im ∂. (The reader may see [9] for similar results in the symplectic case and the proofs are the same in the pseudo-Euclidean case.) For what follows, we fix a decomposition (3.1), which means that we also have E = K ⊕ Q, and ρ| Q is an isomorphism with the inverse σ : T M → Q. We will denote by p K , p Q , p P , p im ∂ , p C the projections of E onto the corresponding subspaces, respectively.
Furthermore, we construct a metric connection ∇ of E as follows. ∇ will be the sum of metric connections of the components P, C. Furthermore, the component ∇ P will be a sum ∇ im ∂ ⊕ ∇ Q where ∇ Q is arbitrary and ∇ im ∂ is defined by the condition ). This condition defines well ∇ im ∂ because g| P is neutral. A metric connection on (E, g) which is obtained by the process described above is said to be suitable, and we fix one such suitable connection.
The component ∇ Q of a suitable connection may be identified with a linear connection D on M by means of the formula The following formula defines a 3-form B 1 ∈ Γ ∧ 3 E * : .
From the fact that the ∇-parallel translations preserve the subbundle im ∂, and formula (2.2), it follows that B 1 satisfies condition (2.10). Hence, formula (2.5) with β = β 1 defined by B 1 yields a structure of a pre-Courant algebroid on (E, g, ρ), with a bracket that we denote by [ , ] 1 . Like for any pre-Courant algebroid, in the transitive case too the brackets [e, ∂f ] are always given by formula (1.11). Furthermore, we get where e ∈ ΓE and we use an arbitrary decomposition (3.1) and an arbitrary suitable connection ∇.
Proof. The first formula follows by applying σ to (2.2), since σ • ρ is the projection p Q and the bracket [ , ] satisfies property i). Furthermore, if we expend the expression of the bracket [ , ] 1 , take the scalar g-product by an arbitrary e ∈ ΓE, and use formula (3.4), we see that (3.5) also holds.
Formulas (3.4) and (3.5) define the projections of the bracket [ , ] 1 on Q and K, hence, completely define the bracket. Q.e.d.
In the following proposition we give a more transparent expression of the bracket [ , ] 1 . Proposition 3.2 Let (E, g, ρ) be a Courant vector bundle with a surjective anchor, for which a choice of a spitting (3.1) and of a suitable connection ∇ is made. Then, (E, g, ρ) has a structure of pre-Courant algebroid with the bracket [ , ] 1 given by the formulas where f, f 1 , f 2 ∈ C ∞ (M); q, q 1 , q 2 ∈ ΓQ; c, c 1 , c 2 ∈ ΓC and γ is defined by (2.4). Furthermore, [ , ] 1 is a Courant algebroid bracket iff the connection ∇ is flat.
Proof. In view of the properties of decomposition (3.1), k ∈ K is completely defined by the scalar products g(k, q) ∀q ∈ Q. But, if e, e 1 , e 2 of (3.5) are in Q, using the definition of a suitable connection we get p K [q 1 , q 2 ] 1 = 0. Accordingly, with (3.4) and (2.2) we get [q 1 , q 2 ] 1 = σ[ρq 1 , ρq 2 ] and the first formula (3.6) follows if we also take into account (1.11).
By similar considerations based on the properties of decomposition (3.1) and of a suitable connection, (3.4), (3.5) and (1.11) yield the remaining formulas (3.6).
From the first formula (3.6), we see that (P, g| P , ρ| P ) (P = Q ⊕ im ∂) is a pre-Courant algebroid with the induced bracket. Moreover, it is easy to check that property iii) of Proposition 1.2 also holds on P , therefore, we actually have a Courant algebroid P , and (3.6) is the structure defined on P ⊕ Q by Proposition 2.14. Accordingly, the last assertion of the present Proposition follows from the last assertion of Proposition 2.14. Q.e.d.
As shown by Proposition 1.4, all the other pre-Courant brackets of the Courant bundle with surjective anchor (E, g, ρ) will be obtained from (3.6) by the addition of a form λ that satisfies the corresponding hypotheses, whence, λ ∈ Γ(∧ 2 E * ⊗ K) and (1.20) holds. The corresponding formulas are In order to get Courant algebroid brackets we must ask λ to satisfy condition (1.17). If expressed on arguments q ∈ ΓQ, c ∈ ΓC, (1.17) decomposes into the following four components In principle, formulas (3.7) and (3.8)-(3.11) yield all the transitive Courant algebroids.
We get a more transparent result if we restrict ourselves to the subclass of transitive Courant algebroids that admit a bracket-closed neutral extension of the subbundle im ∂. We shall call them transitive, restricted, Courant algebroids.
The general bracket (3.7) is restricted iff it is defined by a form λ such that, ∀q 1 , q 2 ∈ ΓQ, λ(q 1 , q 2 ) ∈ im ∂. This condition is equivalent with Λ(q 1 , q 2 , c) = 0, which, because of the skew-symmetry, is equivalent with λ(c, q) ∈ ΓC. Then, we may change the C-component of the connection ∇ by ∇ X C → ∇ X C + λ(c, σ(X)) and get for the same bracket a simplified expression (3.7) that looks as if we have used an additional form λ such that λ(c, q) = 0 and, accordingly, λ(c 1 , c 2 ) ∈ ΓC. Then, conditions (3.8)-(3.11) become simpler and we obtain Proposition 3.3 The bracket (3.7) defines a transitive, restricted, Courant algebroid iff the C-component of the connection ∇ is flat and, in addition to the conditions for a pre-Courant algebroid, the form λ also satisfies the conditions  where λ(q 1 , q 2 ) ∈ im ∂ satisfies the condition (3.13).
Remark 3.2 With the notation used above, let E be a transitive Courant algebroid. Then E/im ∂ gets an induced structure of a transitive Lie algebroid and we have the exact sequence Hence E may be seen as a central extension of a transitive Lie algebroid. In the restricted case, the splitting σ induces a splitting T M σ → E/im ∂ which is a morphism of Lie algebroids. P.Ševera expressed the bracket of the Courant algebroid E by means of the bracket of the Lie algebroid E/im ∂.
The results obtained so far in this section straightforwardly extend to regular Courant algebroids. Indeed, such an algebroid (E, g, ρ) is a transitive Courant algebroid over the base manifold M of E seen as the sum of the leaves of the foliation F = im ρ. Therefore, we get formulas (3.7) again. In order to ensure that we obtain brackets that are differentiable with respect to the original differentiable structure of M it suffices to use metric connections and forms λ, Λ that enjoy this kind of differentiability. Such connections are just Lie algebroid connections for the tangent Lie algebroid of F (called Fpartial connections or connections along the leaves of F in foliation theory). Thus, Propositions 3.2, 3.3 and Corollary 3.1, where we ask the anchor to be surjective over a regular foliation F of M and the connection to be along the leaves of F , describe all the regular (restricted) Courant algebroids.

Miscellanies
a) The basic example of a skew-symmetric Courant algebroid appeared in [1]. It was the vector bundle E = T M ⊕ T * M, endowed with the neutral pseudo-Euclidean metric the non-degenerate cross section ω ∈ ∧ 2 E * , and the projection ρ(X ⊕ α) = X, where X, Y ∈ Γ(T M), α, β ∈ Γ(T * M) and, for clarity, we denoted an element (X, α) = X + α ∈ T M ⊕ T * M by X ⊕ α. Straightforward computations show that the bracket (4.3), together with g, ρ, satisfy the conditions of Definition 1.2, and T M ⊕ T * M is a skewsymmetric Courant algebroid.
For the same data, the formula defines a structure of a non-skew-symmetric Courant algebroid and the Courant bracket (4.3) is the skew-symmetric part of the bracket (4.4). The Courant bracket was extended byŠevera and Weinstein [7] by the addition of a term of the form i(X ∧ Y )Φ to the T * M-component of the right hand side of (4.3), Φ being a closed 3-form on M. The Courant algebroid structures described above are transitive hence, particular cases of the general formulas (3.6), (3.7). Indeed, in the present case, ∂f = 0 ⊕ df (f ∈ C ∞ (M)), im ∂ = T * M, C = {0}, we may take Q = T M, and a suitable connection is provided by any linear connection on M. Then, the first formula (3.6) becomes (4.3). Indeed, this is trivial for α = df 1 , β = df 2 (f 1 , f 2 ∈ C ∞ (M)), and it is true for any α, β because the two formulas (3.6) and (4.3) behave in the same way when arguments are multiplied by a function. The addition of a form λ leads to theŠevera-Weinstein Courant bracket with Φ = −2Λ| Q . Since C = {0}, we necessarily are in the restricted case, λ, Λ satisfy (1.16), and the conditions (3.13)-(3.15) reduce to dΦ = 0. Therefore, the Courant andŠevera-Weinstein brackets define all the Courant algebroid structures on T M ⊕ T * M endowed with the metric (4.1) and the anchor ρ(X ⊕ α) = X.
However, T M ⊕ T * M may have more Courant algebroid structures if, for instance, we change the anchor, as we will see below.
In [4], the Courant bracket was extended to vector bundles A ⊕ A * , where (A, A * ) is a Lie bialgebroid with anchors α, α * , respectively, such that the extended bracket, the metric g defined like in (4.1) and the anchor ρ = α+α * define a structure of skew-symmetric Courant algebroid. The extended bracket is (4.5) [ where d, L, respectively d * , L * , are the exterior differential and Lie derivative associated with the Lie algebroid structure of A, respectively A * , and ω is defined like in (4.2). One can check that the bracket (4.5) is the skew-symmetric part of the product It is possible to connect the bracket (4.5) with a metric connection as indicated in Proposition 2.1, but this doesn't seem to give interesting formulas. On the other hand, we may use the bracket (4.5) and Propositions 1.4, 2.2 in order to derive new Courant algebroid structures on the vector bundles A ⊕ A * and A ⊕ A * ⊕ C for any flat pseudo-Euclidean bundle C.
The Courant bracket (4.3) on T M ⊕ T M * is the particular case of the bracket (4.5) where A = T M with the Lie bracket and A * = T * M with the zero bracket and zero anchor. But, if we assume that P is a Poisson bivector field on M and use the cotangent algebroid structure defined by P on T * M, we get a new Courant algebroid structure on T M ⊕ T M * with the same metric (4.1), with the anchor Id T M ⊕ ♯ P , and with the bracket (4.5). Again, we are in the transitive, restricted case, and it will be possible to express the bracket under the form (3.6). Indeed, the kernel of the anchor is It is easy to see that where X P f is the Hamiltonian vector field of f with respect to P . Then, by technical calculations, one checks that the brackets defined by the first formula (3.6) and by (4.5) coincide; the checks are to be made in each of the cases: two arguments in Q, one in Q and one in im ∂ and two arguments in im ∂.
The original Courant bracket also leads to an example of a non-restrictive, transitive pre-Courant algebroid. Assume that the manifold M is endowed with a Riemannian metric G and consider the vector bundle E = T M ⊕ T * M ⊕ T M with the anchor defined as the projection on the first term and the metric g ⊕ G, where g is given by (4.1). Then, ∂f = 0 ⊕ df ⊕ 0 and E is a Courant vector bundle with surjective anchor, with the natural decomposition E = (Q ⊕ im ∂) ⊕ C where Q = T M and C = (T M, G), and the suitable connection defined by the Levi-Civita connection ∇ of G, which satisfies T ∇,ρ = 0.
Accordingly, we get a pre-Courant algebroid bracket on T M ⊕T * M ⊕T M if we use the corresponding formulas (3.6). The first formula (3.6) again yields the original Courant bracket and the remaining brackets (3.6) are determined by the Levi-Civita connection ∇ of G and the value of γ as defined by (2.4), which yields and X a , Y a ∈ ΓT M, α a ∈ ΓT * M (a = 1, 2). Now, let Φ be a differential 3-form on M and define This form vanishes if one of the arguments belongs to im ∂ and is skew symmetric. Therefore λ of (4.8) may serve as an additional term that leads to a new pre-Courant algebroid bracket (3.7) on T M ⊕ T * M ⊕ T M, which is not restricted. It will be a Courant algebroid bracket iff the conditions (3.8)-(3.11) hold. The significance of these conditions is unclear.
b) The Courant bracket (4.3) may be defined for every Lie algebroid A and it is the particular case of (4.5) where the dual bundle A * is endowed with the zero anchor and the zero bracket. In what follows, we use this remark in order to define a Courant algebroid structure on the tangent bundle of a para-Hermitian manifold M. We recall the definition [2]: a para-Hermitian structure on M consists of a neutral metric g on T M and a (1, 1)-tensor field F that satisfies the conditions (4.10) Condition (4.11), which is the vanishing of the Nijenhuis tensor of F , is the integrability condition of the structure. It follows that M also has a non degenerate 2-form (4.12) ω(X, Y ) = g(F X, Y ), which satisfies the condition and that T M = W + ⊕ W − , where the terms are the ±1-eigendistributions of F and are integrable because of (4.11). We will denote by (4.14) F ± = 1 2 (I ± F ) the projectors on W ± , respectively. From (4.10) and (4.13) it follows that W ± are maximal isotropic subbundles with respect to g and Lagrangian subbundles with respect to ω. Accordingly, the musical isomorphism ♭ g sends W ± onto the dual space W * ∓ and defines an isomorphism T M ≈ W + ⊕ W * + . Because of integrability, W + , with the Lie bracket, is a Lie algebroid and, if we use (4.3) in this case, we get a bracket on the tangent bundle T M defined by (4.15) [ The conclusion is that (T M, g, F + , [ , ] F + ) is a regular, skew-symmetric Courant algebroid. A Courant algebroid structure on T M requires a pseudo-Riemannian metric g on M, and a field φ of endomorphisms of T M, which will be the anchor. Notice that ∂f = (1/2)♯ g (df • φ) and that span{df • φ} = ann ker φ. Hence, φ is a Courant anchor iff im φ is a generalized, completely integrable distribution and ♯ g (ann ker φ) is an isotropic distribution. In particular, we must have In Section 4b) we had an example of such a situation.
Generally, on T M we have the Levi-Civita connection ∇ of g , which is metric, and we may use it to express the bracket. The connection ∇ has no torsion but it has a φ-torsion A technical calculation shows that the φ-torsion is related with the Nijenhuis tensor N φ (see (4.11). Namely, The operator γ of (2.4) will now be defined by and the Courant algebroid bracket will be where β is a tensor field of type (1,2) on M such that φ(β(X, Y )) = T (∇,φ) (X, Y ) and B(X, Y, Z) = g(β(X, Y ), Z) is a 3-form on M.
Thus, essentially, a Courant algebroid structure on (T M, g, φ) is a special type of a 3-form. What we must still ask is the fulfillment of condition C = 0 for C defined by (2.13). Unfortunately, this condition is too complicated and does not provide a practical way to find new Courant algebroids. This is true even if stronger conditions are added. For instance, if we ask φ to be ∇-parallel along paths in the leaves of im φ, (4.17) shows that T (∇,φ) = 0 and we may try the solution β = 0. The resulting bracket (4.20) defines a Courant algebroid structure iff (see (2.12)) Let M be a manifold endowed with a regular foliation F . Let E be a foliated, vector bundle over M endowed with a foliated, pseudo-Euclidean metric g. This means that E has a given, maximal system of local trivializations with transition functions that are constant along the leaves of F (foliated functions), and, ∀e 1 , e 2 ∈ Γ pr E, where Γ pr denotes the space of projectable cross sections of E (i.e., constant along the leaves), the function g(e 1 , e 2 ) is constant along the leaves. (E.g., see [5] for the theory of foliations.) Then, we have Proposition 4.1 On a foliated pair (E, g) there exist metric connections ∇ that satisfy the condition ∇ X e = 0, ∀X ∈ ΓT F and ∀e ∈ Γ pr E.
Proof. Take an arbitrary normal bundle NF (i.e., T M = T F ⊕ NF ) and an arbitrary metric connection∇ of (E, g). Then define It is easy to check that this produces a connection as required. Indeed, we must still define ∇ X e for X ∈ ΓT F and a non projectable e ∈ ΓE. For this purpose, we take local projectable bases e α of E and, for we put Q.e.d. A connection that satisfies the properties stated by Proposition 4.1 will be called an adapted connection of (E, g).
Let (E, g) be foliated and let us assume that there exists a surjective morphism ρ : E → T F , which is a Courant anchor of (E, g). Then, for any adapted connection ∇, one has a non-zero ρ-torsion given by The values of this bracket for arbitrary cross sections f 1 e 1 , f 2 e 2 , e 1 , e 2 ∈ Γ pr E, f 1 , f 2 ∈ C ∞ (M) follows from property iv), Proposition 1.2. Furthermore, the operator γ satisfies the condition C 0 = 0, where C 0 is defined by (2.12); this is obvious for projectable arguments and is true for arbitrary arguments because C 0 is a tensor. Hence, we get Proposition 4.2 For a triple (E, g, ρ) as described above, a Courant algebroid bracket is a bracket of the form (2.5), where β ∈ Γ(∧ 2 E * ⊗ E) is associated with a 3-form B ∈ Γ(∧ 3 E * ) and satisfies the conditions [β(e 1 , e 2 ), e 3 ] 0 , ∀e 1 , e 2 , e 3 ∈ Γ pr E.

Appendix: Dirac linear spaces
The Courant algebroids resulted from the process of studying Dirac structures, which are a significant generalization of the Poisson structures [1]. Although this is not a subject of the present paper, we have added this appendix, which shows that the known linear algebra of Dirac structures is a part of para-Hermitian linear algebra. A para-Hermitian vector space is a 2n-dimensional vector space W that has the structure indicated for the fibers of the tangent bundle of a para-Hermitian manifold in the previous section. On W we have the ingredients g, ω, F, W ± , F ± , with the algebraic properties stated in Section 4b), and W = W + ⊕ W − , ♭ g : W ± ≈ W * ∓ . The space W has adapted bases (b i , c j ) (i, j = 1, ..., n), where (b 1 , ..., b n ) is a basis of W + , therefore, ♭ g b 1 , ..., ♭ g b n is a basis of W * − , and (c 1 , ..., c n ) is the corresponding dual basis of W − , i.e., g(b i , c j ) = δ ij (i, j = 1, ..., n). Proof. The results are analogous to known results for Lagrangian subspaces of a symplectic vector space, and we prove them as in the latter case, e.g., [9]. For any subspace S ⊆ W such that W = L ⊕ S and any basis (l 1 , ..., l n ) of L, there exists a unique conjugated basis (s 1 , ..., s n ) of S, such that g(l i , s j ) = δ ij . Using the conjugated basis, we can obtain vectors u i = s i + τ k i l k (the Einstein summation convention holds) such that g(u i , u j ) = 0, and these vectors span a Dirac subspace L ′ that satisfies L ⊕ L ′ = W . Furthermore, if L ′′ is another Dirac subspace such that L ⊕ L ′′ = W and if (u i ) is the conjugated basis of (l i ) in L ′ and (v i ) is the conjugated basis of (l i ) in L ′′ , there exists a unique skew-symmetric matrix (θ j i ) such that v i = u i + θ j i l j . Q.e.d.
Using adapted bases of W , it follows that the set of the reflectors of W is the n(n − 1)-dimensional homogeneous space R = O(W, g)/pH(W ) ≈ O(n, n)/Gl(n), where O(W, g) is the g-preserving subgroup of the general linear group Gl(W ), which acts transitively on R, pH(W ) is the para-Hermitian subgroup, which commutes with F and is the isotropy subgroup of the pair (W + , W −) ∈ R, O(n, n) ≈ O(W, g) is the subgroup of Gl(2n) which preserves the canonical neutral metric, and Gl(n) ≈ pH(W ) by the embedding (A ∈ Gl(n)) [3,11]. Proposition 5.1 shows that the set D of Dirac subspaces of W is the quotient space of R by the equivalence relation with equivalence classes T (L), hence, D is a [n(n − 1)/]2-dimensional space namely, the homogeneous space O(W, g)/O W + (W, g), where the isotropy group of W + ∈ D at the denominator is that of the elements φ ∈ O(W, g) which satisfy the condition F − •φ•F + = 0.
We also notice that L ⊆ W is a Dirac subspace iff F (L) is the ωorthogonal subspace of L. Therefore, if L is a Dirac subspace, ker(ω| L ) = L ∩ F (L). On the other hand, it follows easily that for a Dirac subspace L one has ker(ω| L ) = (W + ∩ L) ⊕ (W − ∩ L).
The following proposition shows that the integers k = dim(W − ∩ L) and r = rank(ω| L ) are the only invariants of a Dirac subspace with respect to the action of the para-Hermitian subgroup pH(W ).

Proposition 5.2
The group pH(W ) acts transitively on the set of Dirac subspaces L with given values k, r.
Of course, instead of the invariant k = dim(W − ∩ L) we may consider h = dim(W + ∩ L). These two numbers are related by k + h = n − r.