Symmetric brackets induced by connections with totally skew-symmetric torsion on skew-symmetric algebroids
aa r X i v : . [ m a t h . DG ] D ec SYMMETRIC BRACKETS INDUCED BYCONNECTIONS WITH TOTALLY SKEW-SYMMETRIC TORSIONON SKEW-SYMMETRIC ALGEBROIDS
BOGDAN BALCERZAK
Abstract.
In this note, we discuss symmetric brackets on skew-symmetric algebroidsassociated with a metric structure. Given a pseudo-Riemannian metric structure, wedescribe symmetric brackets induced by connections with totally skew-symmetric tor-sion in the language of Lie derivatives and differentials of functions. In particular, weobtain an explicit formula of the Levi-Civita connection. We also present some symmet-ric brackets on almost Hermitian manifolds. Especially, we discuss the first canonicalHermitian connection and show the formula for it in the case of nearly K¨ahler manifoldsusing the properties of symmetric brackets. Introduction An anchored vector bundle ( A, ̺ A ) over a manifold M is a vector bundle A over M equipped with a homomorphism of vector bundles ̺ A : A → T M over the identity, whichis called an anchor . If, additionally, in the space Γ( A ) of smooth sections of A we have R -bilinear skew-symmetric mapping [ · , · ] : Γ( A ) × Γ( A ) → Γ( A ) associated with theanchor with the following derivation law(1.1) [ X, f · Y ] = f · [ X, Y ] + ( ̺ A ◦ X )( f ) · Y for X, Y ∈ Γ( A ), f ∈ C ∞ ( M ), we say that ( A, ρ A , [ · , · ]) is a skew-symmetric algebroid over M .If the anchor preserves [ · , · ] and the Lie bracket [ · , · ] T M of vector fields on M , i.e., ̺ A ◦ [ X, Y ] = [ ̺ A ◦ X, ̺ A ◦ Y ] T M for
X, Y ∈ Γ( A ), a skew-symmetric algebroid is an almostLie algebroid . Any skew-symmetric algebroid in which [ · , · ] satisfies the Jacobi identityis a Lie algebroid in the sense of Pradines, who discovered them as infinitesimal partsof differentiable groupoids [25] (for the general theory of Lie algebroids, we refer to theMackenzie monographs [16], [17]). Thus, Lie algebroids are simultaneous generalizationsof integrable distributions on the one hand, and Lie algebras on the other.Anchored vector bundles, in particular almost Lie algebroids, were intensively studiedby Marcela Popescu and Paul Popescu, among others, in [20], [21], [22], [23] and recentlyin [24], in which the Chern character for almost Lie algebroids is considered. However,the concept of skew-symmetric algebroids was introduced by Kosmann-Schwarzbach andMagri in [13] on the level of finitely generated projective modules over commutative andassociative algebras with unit and under the name pre-Lie algebroids . Skew-symmetricalgebroids (under the same name pre-Lie algebroids) were examined by Grabowski and Mathematics Subject Classification.
Key words and phrases.
Skew-symmetric algebroid; almost Lie algebroid; anchored vector bundle;connection; symmetric product; the symmetrized covariant derivative; symmetric Lie derivative;connection with totally skew-symmetric torsion(
B. Balcerzak ) Institute of Mathematics, Lodz University of Technology, W´olcza´nska 215, 90-924 L´od´z, Poland; e-mail: [email protected].
Urba´nski in [8], [9], where a concept of general algebroids, which play an important rolein analytical mechanics, also was introduced. Using general algebroids instead of Liealgebroids, Grabowska, Grabowski, and Urba´nski observed in [6] and [7] that one candescribe a larger family of systems, both in the Lagrangian and Hamiltonian formalisms.In this paper, we use the terminology of skew-symmetric algebroid which comes fromde Le´on, Marrero, and de Diego in [12], in which linear almost Poisson structures (alsodiscussed [8], [9], [13]) are applied to nonholonomic mechanical systems.Given an anchored vector bundle, we can associate a connection. Given a skew-symmetric algebroid, we can associate a connection with a torsion.An A -connection in a vector bundle E → M is an R -bilinear map ∇ : Γ( A ) × Γ( E ) → Γ( E ) with the following properties: ∇ f · X ( u ) = f · ∇ X ( u ) , ∇ X ( f · u ) = f · ∇ X ( u ) + ( ̺ A ◦ X )( f ) · u for any X, Y ∈ Γ( A ) , f ∈ C ∞ ( M ), u ∈ Γ( E ).The torsion of an A -connection ∇ in A is the tensor T ∇ ∈ Γ( V A ∗ ⊗ A ) defined by T ∇ ( X, Y ) = ∇ X Y − ∇ Y X − [ X, Y ]for
X, Y ∈ Γ( A ). We say that an A -connection is torsion-free if its torsion equals zero.Let S k A ∗ denote the k -th symmetric power of the bundle A ∗ . On S ( A ) = L k ≥ S k A ∗ ,we define the operator d s : S ( A ) → S ( A ) as the symmetrized covariant derivative, i.e., d s η = ( k + 1) · (Sym ◦∇ ) η for η ∈ Γ( S k A ∗ ). This operator appeared naturally in the studyof generalized gradients on Lie algebroids in the sense of Stein-Weiss in [2]. However, theoperator d s in the case of tangent bundles was introduced by Sampson in [26]. Next,this operator on tangent bundles was discussed by several authors when studying theLichnerowicz-type Laplacian on symmetric tensors, called the Sampson Laplacian. Inparticular, we observe an important contribution to the theory of these operators in therecent papers of Mikeˇs, Rovenski, Stepanov, and Tsyganok [27], [19]. We also refer to[19] for the extended literature on the Sampson Laplacian.For η ∈ Γ( S k A ∗ ), X , . . . , X k +1 ∈ Γ ( A ) holds( d s η ) ( X , . . . , X k +1 ) = k +1 P j =1 ( ρ A ◦ X j ) (cid:16) η ( X , . . . b X j . . . , X k +1 ) (cid:17) (1.2) − P i Observe that the symmetric product h· : ·i satisfies the Leibniz rules and ∇ X Y = ([ X, Y ] + h X : Y i ) + T ∇ ( X, Y )for X, Y ∈ Γ ( A ). Thus, the symmetric product induced by ∇ is a summand of theconnection. Our first purpose is to determine the symmetric products for connectionsrelated to the pseudometric structure. We give an explicit formula for a metric connectionwith totally skew-symmetric torsion (but not necessary torsion-free in general) using thelanguage of symmetric product. To describe these symmetric brackets, we use the Liederivative and the exterior derivative operator induced by the structure of the skew-symmetric algebroid and their symmetric counterparts. We show that the condition forconnections with totally skew-symmetric torsion to be compatible with the metric is thatthe (alternating) Lie derivative of the metric is equal to the minus of the symmetric Liederivative of the metric.We also consider an almost Hermitian structure and some symmetric brackets asso-ciated with connections that are compatible with the metric structure and the almostcomplex structure. We consider two structures of the skew-symmetric algebroid in thealmost Hermitian manifold ( M, g, J ). The first structure is the tangent bundle withthe identity as an anchor and with the Lie bracket of vector fields. The second skew-symmetric algebroid structure induced by the almost complex structure J , where J isthe anchor and the bracket is associated with the Nijenhuis tensor, was introduced in[13]. We also discuss the first canonical Hermitian connection ∇ and obtain a formulafor ∇ in the case of nearly K¨ahler manifolds using the properties of symmetric brackets.2. The exterior derivative operator and the symmetrized covariantderivative Let ( A, ̺ A , [ · , · ]) be a skew-symmetric algebroid over a manifold M . The substitution operator i X : Γ( N k A ∗ ) → Γ( N k − A ∗ ) for X ∈ Γ( A ) is defined by( i X ζ )( X , . . . , X k − ) = ζ ( X, X , . . . , X k − )for ζ ∈ Γ( N k A ∗ ), X, X , . . . , X k − ∈ Γ( A ).The ( alternating ) Lie derivative L aX : Γ( N k A ∗ ) → Γ( N k A ∗ ) for X ∈ Γ( A ) is definedby ( L aX Ω) ( X , . . . , X k ) = ( ̺ A ◦ X )(Ω( X , . . . , X k )) − k P i =1 Ω( X , . . . , [ X, X i ] , . . . , X k )for Ω ∈ Γ( N k A ∗ ), X , . . . , X k ∈ Γ( A ). Notice that L aX ( η ) ∈ Γ( V A ∗ ) if η ∈ Γ( V A ∗ ).Moreover, let ∇ : Γ( A ) × Γ( A ) → Γ( A )be an A -connection in A . We define the A -connection ∇ ∗ in the dual bundle in a classicalway by the following formula( ∇ ∗ X ω ) Y = ( ̺ A ◦ X )( ω ( Y )) − ω ( ∇ X Y )for ω ∈ Γ( A ∗ ), X, Y ∈ Γ( A ). Next, by the Leibniz rule, we extend this connection to the A -connection in the whole tensor bundle O A ∗ , which will also be denoted by ∇ . Thenfor ζ ∈ Γ (cid:16)O k A ∗ (cid:17) , X, X , . . . , X k ∈ Γ( A ),( ∇ X ζ )( X , . . . , X k ) = ( ̺ A ◦ X )( ζ ( X , . . . , X k )) − k X j =1 ζ ( X , . . . , ∇ X X j , . . . , X k ) . BOGDAN BALCERZAK Now, we define the operator ∇ : Γ (cid:16)O k A ∗ (cid:17) → Γ (cid:16)O k +1 A ∗ (cid:17) by ( ∇ ζ ) ( X , X . . . , X k +1 ) = ( ∇ X ζ ) ( X , . . . , X k +1 ) . We recall that the exterior derivative operator on the skew-symmetric algebroid( A, ̺ A , [ · , · ]) is defined by( d a η ) ( X , . . . , X k +1 ) = k +1 P j =1 ( − j +1 ( ρ A ◦ X j ) (cid:16) η ( X , . . . b X j . . . , X k +1 ) (cid:17) + P i Jac [ · , · ] : Γ( A ) × Γ( A ) → Γ( A ) of the bracket [ · , · ] given byJac [ · , · ] ( X, Y, Z ) = [[ X, Y ] , Z ] + [[ Z, X ] , Y ] + [[ Y, Z ] , X ]for X, Y, Z ∈ Γ( A ). If the bracket [ · , · ] satisfies the Jacobi identity, i.e., Jac [ · , · ] = 0, d a ◦ d a =0 (discussed in [18]). If ∇ is torsion-free A -connection in A , then d a can be written as thealternation of the operator ∇ (cf. [2]), i.e., d a = ( k + 1) · (Alt ◦∇ ) on Γ( V k A ∗ ), whereAlt is the alternator given by (Alt ζ ) ( X , . . . , X k ) = k ! P σ ∈ S k sgn σ ζ (cid:0) X σ (1) , . . . , X σ ( k ) (cid:1) for ζ ∈ Γ( N k A ∗ ). Equivalently,( d a η ) ( X , . . . , X k +1 ) = k +1 P j =1 ( − j +1 (cid:0) ∇ X j η (cid:1)(cid:16) X , . . . b X j . . . , X k +1 (cid:17) for η ∈ Γ( V k A ∗ ), X , . . . , X k +1 ∈ Γ( A ).Here, we recall the classical Cartan’s formulas: Lemma 2.1. For any X, Y ∈ Γ( A ) , (a) L aX = i X d a + d a i X and (b) L aX i Y − i Y L aX = i [ X,Y ] . The symmetrized covariant derivative is the operator d s = ( k + 1) · (Sym ◦∇ ) : Γ( S k A ∗ ) → Γ( S k +1 A ∗ )being the symmetrization of ∇ up to a constant on the symmetric power bundle, whereSym is the symmetrizer defined by (Sym ζ ) ( X , . . . , X k ) = k ! P σ ∈ S k ζ (cid:0) X σ (1) , . . . , X σ ( k ) (cid:1) for ζ ∈ Γ( N k A ∗ ). Equivalently,(2.1) ( d s η ) ( X , . . . , X k +1 ) = k +1 P j =1 (cid:0) ∇ X j η (cid:1)(cid:16) X , . . . b X j . . . , X k +1 (cid:17) for η ∈ Γ( S k A ∗ ), X , . . . , X k +1 ∈ Γ ( A ). We recall that the operator d s in the case oftangent bundles was introduced by Sampson in [26], in which a symmetric version ofChern’s theorem is proved. This operator on tangent bundles was discussed in [11], inwhich a Fr¨olicher–Nijenhuis bracket for vector valued symmetric tensors is also discussedand in [3], in which the Dirac-type operator on symmetric tensors was considered. One YMMETRIC BRACKETS OF CONNECTIONS WITH TOTALLY SKEW-SYMMETRIC TORSION 5 can check that for η ∈ Γ( S k A ∗ ), X , . . . , X k +1 ∈ Γ ( A ) the following Koszul-type formulaholds ( d s η ) ( X , . . . , X k +1 ) = k +1 P j =1 ( ρ A ◦ X j ) (cid:16) η ( X , . . . b X j . . . , X k +1 ) (cid:17) − P i Let ( A, ̺ A , [ · , · ]) be a skew-symmetric algebroid over a manifold M . A symmetricbracket on the anchored vector bundle ( A, ̺ A ) is an R -bilinear symmetric mapping h· : ·i : Γ( A ) × Γ( A ) → Γ( A )satisfying the following Leibniz-kind rule: h X : f Y i = f h X : Y i + ( ̺ A ◦ X )( f ) Y for X, Y ∈ Γ( A ), f ∈ C ∞ ( M ).Let us assume that the skew-symmetric algebroid ( A, ̺ A , [ · , · ]) is equipped with asymmetric bracket h· : ·i : Γ( A ) × Γ( A ) → Γ( A ).We define the operator d s : Γ( S k A ∗ ) → Γ( S k +1 A ∗ ) on symmetric power bundle S ( A )for η ∈ Γ( S k A ∗ ), X , . . . , X k +1 ∈ Γ ( A ) by( d s η ) ( X , . . . , X k +1 ) = k +1 P j =1 ( ρ A ◦ X j ) (cid:16) η ( X , . . . b X j . . . , X k +1 ) (cid:17) − P i For any X, Y ∈ Γ( A ) , (a) L sX = i X d s − d s i X and (b) L sX i Y − i Y L sX = i h X : Y i .Proof. Let X, Y, X , . . . , X k ∈ Γ( A ) and Ω ∈ Γ( N k A ∗ ).Observe that( i X d s Ω) ( X , . . . , X k )= ( ̺ A ◦ X )(Ω( X , . . . , X k )) + k P i =1 ( ̺ A ◦ X i )(Ω( X, X , . . . , b X i , . . . , X k )) − k P i =1 Ω( X , . . . , h X : X i i , . . . , X k ) − P i YMMETRIC BRACKETS OF CONNECTIONS WITH TOTALLY SKEW-SYMMETRIC TORSION 7 Lemma 3.2. For f ∈ C ∞ ( M ) , X ∈ Γ( A ) , ω ∈ Γ( A ∗ ) , we have (a) L sf · X ω = f · L sX ω − ( i X ω ) · d s f and (b) L sX ( f · ω ) = f · L sX ω + ( ̺ A ◦ X )( f ) · ω . The symmetric bracket of a skew-symmetric algebroid withtotally skew-symmetric torsion Let ( A, ̺ A , [ · , · ]) be a skew-symmetric algebroid over a manifold M equipped with apseudo-Riemannian metric g ∈ Γ( S A ∗ ) in the vector bundle A and an A -connection ∇ in A . Let h· : ·i be the symmetric product induced by ∇ and d s the symmetrized covariantderivative. The pseudo-Riemannian metric defines two homomorphisms of vector bundles ♭ : A → A ∗ ,♯ : A ∗ −→ A by ♭ ( X ) = i X g and g ( ♯ ( ω ) , X ) = ω ( X )for X ∈ Γ( A ), ω ∈ Γ ( A ∗ ), respectively. For any X ∈ Γ( A ), the 1-form i X g = g ( X, · ) willbe denoted, briefly, by X ♭ .We say that ∇ is a connection with totally skew-symmetric torsion with respect to apseudo-Riemannian metric g if the tensor T g ∈ Γ (cid:16)O A ∗ (cid:17) given by T g ( X, Y, Z ) = g ( T ∇ ( X, Y ) , Z )for X, Y, Z ∈ Γ( A ), is a 3-form on A , i.e., T g ∈ Γ( V A ∗ ) (cf. [1]). Theorem 4.1. Let X, Z ∈ Γ( A ) .Then g ( ∇ X X, Z ) = g ( ♯ ( L aX X ♭ − d a ( g ( X, X )) , Z ) − g ( T ∇ ( X, Z ) , X )+( ∇ g )( Z, X, X ) − ( d s g )( X, X, Z ) . In particular, if ∇ is a connection with totally skew-symmetric torsion compatible with g , then (4.1) ∇ X X = ♯ ( L aX X ♭ − d a ( g ( X, X )) . Proof. Let X, Z ∈ Γ( A ). First, observe that( d s g )( X, X, Z ) = 2( ∇ g )( X, X, Z ) + ( ∇ g )( Z, X, X ) . Therefore, we have( ∇ g )( Z, X, X ) − ( d s g )( X, X, Z ) = ( ∇ g )( Z, X, X ) − ( ∇ g )( X, X, Z ) . Next, observe that ( ∇ g )( Z, X, X ) − ( ∇ g )( X, X, Z )= ( ∇ Z g )( X, X ) − ( ∇ X g )( X, Z )= ̺ A ( Z )( g ( X, X )) − g ( ∇ Z X, X ) − ̺ A ( X )( g ( X, Z )) + g ( ∇ X X, Z ) + g ( X, ∇ X Z )= ̺ A ( Z )( g ( X, X )) + g ( ∇ X Z − ∇ Z X − [ X, Z ] , X ) − ̺ A ( X )( g ( X, Z )) + g ([ X, Z ] , X ) + g ( ∇ X X, Z ) . BOGDAN BALCERZAK Since ̺ A ( Z )( g ( X, X )) = d a ( g ( X, X ))( Z ) = g ( ♯ ( d a ( g ( X, X ))) , Z )and (cid:0) L aX X ♭ (cid:1) ( Z ) = ̺ A ( X )( X ♭ ( Z )) − X ♭ ([ X, Z ])= ̺ A ( X )( g ( X, Z )) − g ( X, [ X, Z ]) , we have ( ∇ g )( Z, X, X ) − ( ∇ g )( X, X, Z )= d a ( g ( X, X ))( Z ) + g ( T ∇ ( X, Z ) , X ) − (cid:0) L aX X ♭ (cid:1) ( Z ) + g ( ∇ X X, Z ) . Moreover, if ∇ is a metric connection with totally skew-symmetric torsion, then ∇ g = 0, d s g = 0, and g ( T ∇ ( X, Z ) , X ) = − g ( T ∇ ( X, X ) , Z ) = 0 , and, in consequence, we obtain (4.1). This completes the proof. (cid:3) Applying Theorem 4.1, we have Theorem 4.2. Let X, Y, Z ∈ Γ( A ) and let h X : Y i be the symmetric bracket of sectionsinduced by ∇ , i.e., h X : Y i = ∇ X Y + ∇ Y X . Then g ( h X : Y i , Z ) = g ( ♯ ( L aX Y ♭ + L aY X ♭ − d a ( g ( X, Y ))) , Z )(4.2) − g ( T ∇ ( X, Z ) , Y ) − g ( T ∇ ( Y, Z ) , X )+2( ∇ g )( Z, X, Y ) − ( d s g )( X, Y, Z ) . Proof. Using the following polarization formula h X : Y i = ∇ X + Y ( X + Y ) − ∇ X X − ∇ Y Y and Theorem 4.1, we obtain h X : Y i = g ( ♯ ( L aX + Y ( X + Y ) ♭ − d a ( g ( X + Y, X + Y )) , Z ) − g ( T ∇ ( X + Y, Z ) , X + Y )+( ∇ g )( Z, X + Y, X + Y ) − ( d s g )( X + Y, X + Y, Z ) − g ( ♯ ( L aX X ♭ − d a ( g ( X, X )) , Z )+ g ( T ∇ ( X, Z ) , X ) − ( ∇ g )( Z, X, X ) + ( d s g )( X, X, Z ) − g ( ♯ ( L aY Y ♭ − d a ( g ( Y, Y ))+ g ( T ∇ ( Y, Z ) , Y ) − ( ∇ g )( Z, Y, Y ) + ( d s g )( Y, Y, Z ) . First observe that L aX + Y ( X + Y ) ♭ − L aX X ♭ − L aY Y ♭ = L aX Y ♭ + L aY X ♭ and − d a ( g ( X + Y, X + Y ) + d a ( g ( X, X )) + d a ( g ( Y, Y ) = − d a ( g ( X, Y )) . Since g is a symmetric tensor and T ∇ is skew-symmetric, we conclude that − g ( T ∇ ( X + Y, Z ) , X + Y ) + g ( T ∇ ( X, Z ) , X ) + g ( T ∇ ( Y, Z ) , Y )is equal to − g ( T ∇ ( X, Z ) , Y ) − g ( T ∇ ( Y, Z ) , X ) . Moreover,( ∇ g )( Z, X + Y, X + Y ) − ( ∇ g )( Z, X, X ) − ( ∇ g )( Z, Y, Y ) = 2( ∇ g )( Z, X, Y ) YMMETRIC BRACKETS OF CONNECTIONS WITH TOTALLY SKEW-SYMMETRIC TORSION 9 and ( d s g )( X, Y, Z ) = ( d s g )( X, Y, Z ) + ( d s g )( Y, X, Z )= ( d s g )( X + Y, X + Y, Z ) − ( d s g )( X, X, Z ) − ( d s g )( Y, Y, Z ) . Hence, it is clear that some summands of h X : Y i cancel. This establishes the formula(4.2). (cid:3) The formula in Theorem 4.2 gives an explicit formula of symmetric bracket definedby any metric connection with totally skew-symmetric torsion. Corollary 4.3. Let ∇ be any metric A -connection in A with totally skew-symmetrictorsion with respect to a pseudo-Riemannian metric g . Then ∇ X Y + ∇ Y X = ♯ ( L aX Y ♭ + L aY X ♭ − d a ( g ( X, Y )) . A general metric compatibility condition of connections havingtotally skew-symmetric torsion. The Levi-Civita connection Let ( A, ̺ A , [ · , · ]) be a skew-symmetric algebroid over a manifold M equipped with apseudo-Riemannian metric g ∈ Γ( S A ∗ ) in the vector bundle A and a symmetric bracket h· : ·i : Γ( A ) × Γ( A ) → Γ( A ). By definition, we recall that the symmetric bracket is an R -bilinear symmetric mapping which satisfies the following Leibniz-kind rule: h X : f Y i = f h X : Y i + ( ̺ A ◦ X )( f ) · Y for X, Y ∈ Γ( A ), f ∈ C ∞ ( M ).Given the bundle metric g on A , there is a unique A -connection in A which is torsion-free and metric-compatible (i.e., T ∇ = 0 and ∇ g = 0). We call such an A -connectionthe Levi-Civita connection with respect to g . Let L s and d s denote the symmetric Liederivative and the symmetric derivative operator, respectively, and both are induced by h· : ·i . Theorem 5.1. Let ∇ be an A -connection in A with totally skew-symmetric torsion withrespect to a pseudo-Riemannian metric g on A given by (5.1) ∇ X Y = ([ X, Y ] + h X : Y i ) + T ( X, Y ) for X, Y ∈ Γ( A ) . Then ( i X ◦ ∇ ) g = ( L aX + L sX ) g for X ∈ Γ( A ) .Proof. Let X, Y, Z ∈ Γ( A ). Since T ∈ Γ (cid:16)^ A ∗ ⊗ A (cid:17) is a 2-skew-symmetric tensor withthe property that g ( Y, T ( X, Z )) = g ( T ( X, Z ) , Y ) = − g ( T ( X, Y ) , Z ) , we have,( ∇ X g ) ( Y, Z ) = ρ A ( X )( g ( Y, Z )) − g ( ∇ X Y, Z ) − g ( Y, ∇ X Z )= ( ρ A ( X )( g ( Y, Z )) − g ([ X, Y ] , Z ) − g ( Y, [ X, Z ]))+ ( ρ A ( X )( g ( Y, Z )) − g ( h X : Y i , Z ) − g ( Y, h X : Z i )) − g ( T ( X, Y ) , Z ) − g ( Y, T ( X, Z ))= ( L aX g + L sX g ) ( Y, Z ) + 0 . (cid:3) Hence, we can conclude the following condition on a connection with totally skew-symmetric torsion to be a metric connection: Corollary 5.2. If ∇ is an A -connection with totally skew-symmetric torsion with respectto g given by (5.1) , then ∇ is metric with respect to g if and only if L aX g = −L sX g for any X ∈ Γ( A ) . Now, we recall some properties of the (skew-symmetric) Lie derivative. Lemma 5.3. For f ∈ C ∞ ( M ) , X ∈ Γ( A ) , ω ∈ Γ( A ∗ ) , we have (a) L af · X ω = f · L aX ω + ( i X ω ) · d a f and (b) L aX ( f · ω ) = f · L aX ω + ( ̺ A ◦ X )( f ) · ω . Theorem 5.4. Given a skew-symmetric algebroid ( A, ̺ A , [ · , · ]) , we define h X : Y i s : Γ( A ) × Γ( A ) → Γ( A ) by (5.2) h X : Y i s = ♯ ( L aX Y ♭ + L aY X ♭ − d a ( g ( X, Y )) for X, Y ∈ Γ( A ) . Then, h· : ·i s is a symmetric bracket that defines the symmetric Liederivative L s satisfying L sX g = −L aX g .Proof. It is evident that h· : ·i s is a symmetric and R -bilinear mapping. Let X, Y, Z ∈ Γ( A ). Lemma 5.3 now gives L aX ( f Y ) ♭ = f L aX Y ♭ + ( ̺ A ◦ X )( f ) Y ♭ and L afY X ♭ = f L aY X ♭ + g ( X, Y ) d a f. Since d a ( g ( X, f Y )) = f d a ( g ( X, Y )) + g ( X, Y ) d a f, we conclude that h· : ·i s satisfies the Leibniz rule. In consequence, h· : ·i s is a symmetricbracket. Observe that g ( h X : Y i s , Z ) = ( h X : Y i s ) ♭ ( Z ) = ( L aX Y ♭ + L aY X ♭ − d a ( g ( X, Y ))( Z )= ( ̺ A ◦ X )( g ( Y, Z )) − g ( Y, [ X, Z ])+( ̺ A ◦ Y )( g ( X, Z )) − g ( X, [ Y, Z ]) − ( ̺ A ◦ Z )( g ( X, Y )) . Similarly, g ( Y, h X : Z i s ) = ( ̺ A ◦ X )( g ( Y, Z )) − g ( Z, [ X, Y ])+( ̺ A ◦ Z )( g ( X, Y )) − g ( X, [ Z, Y ]) − ( ̺ A ◦ Y )( g ( X, Z )) . Therefore,( L sX g )( Y, Z ) = ( ̺ A ◦ X )( g ( Y, Z )) − g ( h X : Y i s , Z ) − g ( Y, h X : Z i s )= g ( Y, [ X, Z ]) + g ( X, [ Y, Z ]) − ( ̺ A ◦ X )( g ( Y, Z )) + g ( Z, [ X, Y ]) + g ( X, [ Z, Y ])= − ( ̺ A ◦ X )( g ( Y, Z )) + g ([ X, Y ] , Z ) + g ( Y, [ X, Z ])+ g ( X, [ Y, Z ] + [ Z, Y ])= − ( L aX g )( Y, Z ) + 0 . (cid:3) YMMETRIC BRACKETS OF CONNECTIONS WITH TOTALLY SKEW-SYMMETRIC TORSION 11 Theorem 5.1 now yields Corollary 5.5. If ∇ is an A -connection in the bundle A with totally skew-symmetrictorsion defined, for X, Y ∈ Γ( A ) , by ∇ X Y = ([ X, Y ] + h X : Y i s ) + T ( X, Y ) , where h X : Y i s is given in (5.2) , then ∇ is compatible with the metric g . In particular, we can write the Levi-Civita connection explicitly: Corollary 5.6. The Levi-Civita connection with respect to g is given by ∇ X Y = ([ X, Y ] + h X : Y i s ) , where (5.3) h X : Y i s = ♯ ( L aX Y ♭ + L aY X ♭ − d a ( g ( X, Y )) for X, Y ∈ Γ( A ) . Theorem 5.7. The mapping {· , ·} s : Γ( A ) × Γ( A ) → Γ( A ) defined by (5.4) { X, Y } s = ♯ ( L sX Y ♭ + L sY X ♭ + d s ( g ( X, Y )) for X, Y ∈ Γ( A ) is a symmetric bracket in the skew-symmetric algebroid ( A, ̺ A , [ · , · ]) .Proof. It is obvious that {· , ·} s is a symmetric bilinear mapping over R . Let X, Y ∈ Γ( A )and f ∈ C ∞ ( M ). On account of the properties of the symmetric Lie derivatives writtenin Lemma 3.2, L sX ( f Y ) ♭ = f L sX ( Y ♭ ) + ( ̺ A ◦ X )( f ) Y ♭ and L sfY X ♭ = f L sY X ♭ − ( i Y X ♭ ) d s f = f L sY X ♭ − g ( X, Y ) d s f. Furthermore, since d s ( g ( X, f Y )) = f d s ( g ( X, Y )) + g ( X, Y ) d s f, we immediately conclude that { X, f Y } s = f { X, Y } s + ♯ (( ̺ A ◦ X )( f ) Y ♭ ) = f { X, Y } + ( ̺ A ◦ X )( f ) Y, and thus, the proof is complete. (cid:3) To compare the symmetric brackets h· : ·i s and {· , ·} s given in (5.3) and (5.4), respec-tively, we note that h· : ·i s is a symmetric product induced by the Levi-Civita connection,and then, for any X, Y ∈ Γ( A ), we have L sX Y ♭ = L sX i Y g = i Y L sX g + i h X : Y i g = − i Y L aX g + i h X : Y i g = −L aX Y ♭ + i [ X,Y ] g + i h X : Y i g since Theorem 5.1 and the Cartan identities for Lie derivatives given in lemmas 2.1 and3.1 hold. It follows that { X, Y } s = 2 h X : Y i − h X : Y i s for X, Y ∈ Γ( A ). Note that there is a more general property saying that the affine sumof symmetric brackets is again a symmetric bracket. Symmetric brackets on almost Hermitian manifolds In this section we consider the symmetric brackets induced by the structures of almostHermitian manifolds. Let ( M, g, J ) be an almost Hermitian manifold, i.e., ( M, g ) is a 2 n -dimensional Riemannian manifold admitting an orthogonal almost complex structure J : T M → T M . Associated to the structures g and J are the K¨ahler form Ω ∈ Γ( V T ∗ M )given by Ω( X, Y ) = g ( J X, Y )for X, Y ∈ Γ( T M ) and the Nijenhuis tensor N J ∈ Γ( V T ∗ M ⊗ T M ) of J , which isdefined by N J ( X, Y ) = J [ J X, Y ] + J [ X, J Y ] + [ X, Y ] − [ J X, J Y ]for X, Y ∈ Γ( T M ).Kosmann-Schwarzbach and Magri introduced in [13] (cf. also [8]) the bracket [[ · , · ]] J on T M defined by(6.1) [[ X, Y ]] J = [ J X, Y ] + [ X, J Y ] − J [ X, Y ] . One can observe that for any X, Y ∈ Γ( T M ) we have N J ( X, Y ) = J [[ X, Y ]] J − [ J X, J Y ] . Since [[ X, f Y ]] J = f [[ X, Y ]] J + ( J X )( f ) Y for X, Y ∈ Γ( T M ) and f ∈ C ∞ ( M ), the tangent bundle together with the almostcomplex structure J as an anchor and the mapping [[ · , · ]] J given in (6.1) as a skew-symmetric bracket is a skew-symmetric algebroid, which we denote by T M J . It is ob-vious that if N J = 0, then [[[[ X, Y ]] J , Z ]] J = − J [[ J X, J Y ] , J Z ] for X, Y, Z ∈ Γ( T M )and so Jac [[ · , · ]] J ( X, Y, Z ) = − J Jac [ · , · ] ( J X, J Y, J Z ) = 0 for X, Y, Z ∈ Γ( T M ). In conse-quence, if the almost complex structure J is integrable, then the skew-symmetric alge-broid (cid:0) T M, J, [[ · , · ]] J (cid:1) is a Lie algebroid over M .Let d a and L a be the exterior derivative operator and the (alternating) Lie derivativefor the Lie algebroid ( T M, Id T M , [ · , · ]), respectively, where [ · , · ] is the Lie bracket ofvector fields on M . However, let d J and L J be the exterior derivative operator and theLie derivative for the skew-symmetric algebroid (cid:0) T M, J, [[ · , · ]] J (cid:1) , respectively.Using the form of symmetric brackets induced by connections with totally skew-symmetric torsion, in particular by the Levi-Civita connections (Corollary 5.6), we com-pare symmetric brackets induced by the Levi-Civita connections in skew-symmetric al-gebroids T M and T M J , obtaining the following theorem. Theorem 6.1. Let ∇ g be the Levi-Civita connection in the Lie algebroid ( T M, Id T M , [ · , · ]) and let ∇ J,g be the Levi-Civita connection in the skew-symmetric algebroid (cid:0) T M, J, [[ · , · ]] J (cid:1) ,both metric with respect to g . If h X : Y i = ∇ gX Y + ∇ gY X and h X : Y i J = ∇ J,gX Y + ∇ J,gY X for X, Y ∈ Γ( T M ) , then (6.2) h X : Y i J = h J X : Y i + h X : J Y i + ♯ (cid:16) h X : Y i ♭ ◦ J (cid:17) . Proof. Let X, Y ∈ Γ( T M ). Corollary 5.6 now yields h X : Y i J = ♯ ( L JX Y ♭ + L JY X ♭ − d J ( g ( X, Y )) . One can check that L JX Y ♭ = L aJX Y ♭ + L aX ( J Y ) ♭ + ( L aX Y ♭ ) ◦ J. YMMETRIC BRACKETS OF CONNECTIONS WITH TOTALLY SKEW-SYMMETRIC TORSION 13 Moreover, since d J ( g ( X, Y )) = d a ( g ( X, Y )) ◦ J and d a ( g ( J X, Y )) + d a ( g ( X, J Y )) = 0 , we have (6.2). (cid:3) Now, we define some symmetric brackets on almost Hermitian manifolds.Let ( T M, ρ, [ · , · ] ρ ) be a structure of skew-symmetric algebroid, and let h· : ·i ρ be asymmetric bracket in this algebroid. By definition, h X : f Y i ρ = f h X : Y i ρ + ( ρ ◦ X )( f ) Y for X, Y ∈ Γ( T M ).We define two R -bilinear symmetric operators P ρ , Q ρ : Γ( T M ) × Γ( T M ) → Γ( T M ), P ρ ( X, Y ) = − J ([ X, J Y ] ρ + [ Y, J X ] ρ )and Q ρ ( X, Y ) = − J ( h X : J Y i ρ + h Y : J X i ρ )for X, Y ∈ Γ( T M ). Lemma 6.2. For any X, Y ∈ Γ( T M ) , f ∈ C ∞ ( M ) , we have (a) P ρ ( X, f · Y ) = f · P ρ ( X, Y ) + ( ρ ◦ X )( f ) · Y + ( ρ ◦ J X )( f ) · J Y and (b) Q ρ ( X, f · Y ) = f · Q ρ ( X, Y ) + ( ρ ◦ X )( f ) · Y − ( ρ ◦ J X )( f ) · J Y. Proof. Compute directly, P ρ ( X, f · Y ) = − J ([ X, f · J Y ] ρ + [ f · Y, J X ] ρ )= − J ( f · [ X, J Y ] ρ + ( ρ ◦ X )( f ) · J Y + f · [ Y, J X ] ρ − ( ρ ◦ J X )( f ) · Y )= f · P ρ ( X, Y ) + ( ρ ◦ X )( f ) · Y + ( ρ ◦ J X )( f ) · J Y and Q ρ ( X, f · Y ) = − J ( h X : f · J Y i ρ + h f · Y : J X i ρ )= − J ( f · h X : J Y i ρ + ( ρ ◦ X )( f ) · J Y + f · h Y : J X i ρ + ( ρ ◦ J X )( f ) · Y )= f · Q ρ ( X, Y ) + ( ρ ◦ X )( f ) · Y − ( ρ ◦ J X )( f ) · J Y. (cid:3) In consequence of Lemma 6.2, we immediately get the following results. Theorem 6.3. The mapping ( P ρ + Q ρ ) is a symmetric bracket in the skew-symmetric algebroid ( T M, ρ, [ · , · ] ρ ) . Corollary 6.4. The mapping h· : ·i : Γ( T M ) × Γ( T M ) → Γ( T M ) given by h X : Y i = − J (cid:0) [ X, J Y ] + [ Y, J X ] + ♯ ( L aX ( J Y ) ♭ + L aY ( J X ) ♭ + L aJX Y ♭ + L aJY X ♭ ) (cid:1) is a symmetric bracket in the Lie algebroid ( T M, Id T M , [ · , · ]) , where [ · , · ] is the Lie bracketof vector fields on M and L a is the Lie derivative on M . Proof. Let d a be the exterior derivative on manifold M . Taking ρ = Id T M in Theorem6.3 and using Theorem 5.4, we deduce that the formula h X : Y i = − J ([ X, J Y ] + [ Y, J X ]) − ( J ◦ ♯ )( L aX ( J Y ) ♭ + L aJY X ♭ − d a ( g ( X, J Y )) − ( J ◦ ♯ )( L aJX Y ♭ + L aY ( J X ) ♭ − d a ( g ( J X, Y ))defines a symmetric bracket in the tangent bundle with Id T M as an anchor and with theclassical Lie bracket. Since Ω is a skew-symmetric 2-form on M , it follows that g ( X, J Y ) + g ( J X, Y ) = Ω( Y, X ) + Ω( Y, X ) = 0 . Therefore, h X : Y i = − J ([ X, J Y ] + [ Y, J X ]) − ( J ◦ ♯ )( L aX ( J Y ) ♭ + L aY ( J X ) ♭ + L aJX Y ♭ + L aJY X ♭ ) . (cid:3) It is obvious that the bracket in Corollary 6.4 is a totally symmetric part of theconnection ∇ J : Γ( T M ) × Γ( T M ) → Γ( T M ) defined by ∇ JX Y = − J ([ X, J Y ] + h X : J Y i ) . Let ∇ be the Levi-Civita connection with respect to g given by ∇ X Y = (cid:16) [ X, Y ] + h X : Y i ∇ (cid:17) . Now, let h· : ··i = h· : ··i ∇ . Hence, ∇ JX Y = − J ∇ X ( J Y ) . One can observe that the affine sum ∇ = (cid:0) ∇ + ∇ J (cid:1) of connections ∇ and ∇ J is Lichnerowicz’s first canonical Hermitian connection [15],which is compatible with both the metric structure and the almost complex structure.This is a direct consequence of the properties of ∇ and ∇ J given in the following lemma. Lemma 6.5. (a) ( ∇ J g )( X, Y, Z ) = ( ∇ g )( X, J Y, J Z ) for X, Y, Z ∈ Γ( T M ) and (b) ∇ J J = −∇ J . Lemma 6.5 now yields ∇ J g = 0 , ∇ = (cid:0) ∇ + ∇ J (cid:1) g = (cid:0) ∇ g + ∇ J g (cid:1) = 0and ∇ J = (cid:0) ∇ J + ∇ J J (cid:1) = ( ∇ J − ∇ J ) = 0 . We will now consider some further properties of ∇ J and ∇ .For an A -connection ∇ on A , we define the operators d a ∇ , d s ∇ : Γ( N k T ∗ M ) → Γ( N k +1 T ∗ M )as the alternation and the symmetrization of ∇ , respectively, i.e., for ζ ∈ Γ( N k T ∗ M ), X , . . . , X k +1 ∈ Γ( T M ), we have( d a ∇ ζ )( X , . . . , X k +1 ) = k +1 P i =1 ( − i +1 ( ∇ X i ζ ) ( X , . . . b X i . . . , X k +1 ) YMMETRIC BRACKETS OF CONNECTIONS WITH TOTALLY SKEW-SYMMETRIC TORSION 15 and ( d s ∇ ζ )( X , . . . , X k +1 ) = k +1 P i =1 ( ∇ X i ζ ) ( X , . . . b X i . . . , X k +1 ) . We say that an almost Hermitian manifold ( M, g, J ) is nearly K¨ahler if ( ∇ X J ) Y = − ( ∇ Y J ) Y for X, Y ∈ Γ( T M ) (cf. [10]). We have the following lemma. Lemma 6.6. An almost manifold ( M, g, J ) is nearly K¨ahler if and only if d s ∇ J = 0 . Moreover, if ( M, g, J ) is nearly K¨ahler, ∇ is a Hermitian connection with totallyskew-symmetric torsion (e.g., cf. [1]).Now, we compare the symmetric brackets induced by ∇ and ∇ . We will denote by h· : ·i ∇ the symmetric product of ∇ . Theorem 6.7. For X, Y ∈ Γ( T M ) , we have J (( d s ∇ J )( X, Y )) = h X : Y i ∇ − h X : Y i ∇ J . Proof. We first observe that( d s ∇ J )( X, Y ) = ( ∇ X J ) Y + ( ∇ Y J ) X = ∇ X ( J Y ) + ∇ Y ( J X ) − J ( ∇ X Y + ∇ Y X )= ∇ X ( J Y ) + ∇ Y ( J X ) − J h X : Y i ∇ . From this equality, we obtain J (( d s ∇ J )( X, Y )) = J ∇ X ( J Y ) + J ∇ Y ( J X ) + h X : Y i ∇ = − h X : Y i ∇ J + h X : Y i ∇ . (cid:3) Theorem 6.8. For X, Y ∈ Γ( T M ) , we have h X : Y i ∇ = h X : Y i ∇ − J (( d s ∇ J )( X, Y )) . Proof. Since ∇ = (cid:0) ∇ + ∇ J (cid:1) is an affine sum of connections ∇ and ∇ J , h X : Y i ∇ = h X : Y i ∇ + h X : Y i ∇ J . From this result and Theorem 6.7, we see that h X : Y i ∇ = h X : Y i ∇ + (cid:16) h X : Y i ∇ − J (( d s ∇ J )( X, Y )) (cid:17) = h X : Y i ∇ − J (( d s ∇ J )( X, Y )) . (cid:3) Since ∇ = (cid:0) ∇ + ∇ J (cid:1) and ∇ is torsion-free, we have T ∇ = T ∇ + T ∇ J = T ∇ J . Theorem 6.9. T ∇ J = − J ◦ ( d a ∇ J ) .Proof. Let X, Y ∈ Γ( T M ). Then( d a ∇ J )( X, Y ) = ( ∇ X J ) Y − ( ∇ Y J ) X = ∇ X ( J Y ) − ∇ Y ( J X ) − J [ X, Y ] . Hence, − J (( d a ∇ J )( X, Y )) = − J ∇ X ( J Y ) − ( − J ∇ Y ( J X )) + J [ X, Y ]= ∇ JX Y − ∇ JY X − [ X, Y ] = T ∇ J ( X, Y ) . (cid:3) Theorem 6.10. For X, Y ∈ Γ( T M ) , we have T ∇ J ( X, Y ) = − N J ( X, Y ) + ( d s ∇ J )( X, J Y ) − ( d s ∇ J )( J X, Y ) . In particular, if ( M, g, J ) is nearly K¨ahler, then T ∇ J = − N J . Proof. Let X, Y ∈ Γ( T M ). Then (e.g., [1] shows the first equality): − N J ( X, Y ) = ( ∇ X J ) J Y − ( ∇ Y J ) J X + ( ∇ JX J ) Y − ( ∇ JY J ) X = ( ∇ X J ) J Y − ( ∇ Y J ) J X − ( ∇ Y J ) J X + ( d s ∇ J )( J X, Y )+( ∇ X J ) J Y − ( d s ∇ J )( X, J Y )= 2 (( ∇ X J ) J Y − ( ∇ Y J ) J X ) + ( d s ∇ J )( J X, Y ) − ( d s ∇ J )( X, J Y ) . Moreover,( ∇ X J ) J Y − ( ∇ Y J ) J X = −∇ X Y − J ( ∇ X ( J Y )) + ∇ Y X + J ( ∇ Y ( J X ))= − J ( ∇ X ( J Y )) − ( − J ( ∇ Y ( J X ))) − ∇ X Y + ∇ Y X = ∇ JX Y − ∇ JY X − [ X, Y ] = T ∇ J ( X, Y ) . It follows that − N J ( X, Y ) = 2 T ∇ J ( X, Y ) + ( d s ∇ J )( J X, Y ) − ( d s ∇ J )( X, J Y ) . (cid:3) Since ∇ is a totally skew-symmetric connection, Theorem 6.8 now leads to ∇ X Y = (cid:16) [ X, Y ] + h X : Y i ∇ (cid:17) + T ∇ ( X, Y )(6.3) = (cid:16) [ X, Y ] + h X : Y i ∇ − J (( d s ∇ J )( X, Y )) (cid:17) + T ∇ ( X, Y )= ∇ X Y − J (( d s ∇ J )( X, Y )) + T ∇ J ( X, Y ) . 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