Foliations induced by metallic structures
aa r X i v : . [ m a t h . DG ] M a r Foliations induced by metallic structures
Adara M. Blaga and Antonella Nannicini
Abstract
We give necessary and sufficient conditions for the real distributions defined bya metallic pseudo-Riemannian structure to be integrable and geodesically invari-ant, in terms of associated tensor fields to the metallic structures and of adaptedconnections. In the integrable case, we prove a Chen-type inequality for these dis-tributions and provide conditions for a metallic map to preserve these distributions.If the structure is metallic Norden, we describe the complex metallic distributionsin the same spirit.
Let M be a smooth manifold and let J be a (1 , M . If J = pJ + qI ,for some p and q real numbers, then J is called a metallic structure on M and ( M, J )is called a metallic manifold . If g is a pseudo-Riemannian metric on M such that J is g -symmetric, then ( J, g ) is called a metallic pseudo-Riemannian structure on M .The aim of this paper is to consider the complementary distributions associated toa metallic pseudo-Riemannian structure and study their integrability and geodesicallyinvariance in terms of associated tensor fields to the metallic structure and of adaptedconnections. In this sense, we study Schouten-van Kampen, Vr˘anceanu and Vidal con-nections, which seem to be the most important connections for the study of foliations of apseudo-Riemannian manifold [1]. Moreover, for these distributions, we prove a Chen-typeinequality giving a relation between the squared norm of the mean curvature and theChen first invariant. We also prove a leaf correspondence theorem between the leaves oftwo metallic pseudo-Riemannian manifolds when we have a metallic map between themwith certain properties. Mathematics Subject Classification . 53B05, 53C12, 53C15.
Key words and phrases . foliated manifold; metallic pseudo-Riemannian structure. oliations induced by metallic structures p + 4 q is very important in the study of foliations induced by metallicstructures because if it is positive, then J has two real eigenvalues and if it is negative, J has two complex eigenvalues. In the real case, J can be related to almost productstructures and in the complex case, to Norden structures on ( M, g ). In this paper weconsider both of these cases and we describe some similarities and differences betweenthem. In particular, in the complex case, we compute the ¯ ∂ -operator in terms of J .Moreover, we construct the metallic complex cohomology and homology groups.Remark that some properties of metallic distributions have also been studied in [11]. Definition . [3] Let ( M, g ) be a pseudo-Riemannian manifold and let J be ametallic structure on M . We say that the pair ( J, g ) is a metallic pseudo-Riemannianstructure on M if J is g -symmetric. In this case, ( M, J, g ) is called a metallic pseudo-Riemannian manifold . If p + 4 q <
0, then (
J, g ) is called a metallic Norden structure and (
M, J, g ) is called a metallic Norden manifold . Remark . Let (
M, g ) be a pseudo-Riemannian manifold and let J be a metallicstructure on M such that J = pJ + qI . If we require that J is g -skew-symmetric,then we obtain that p = 0. Namely, if we assume g ( J X, Y ) = − g ( X, J Y ), for any X , Y ∈ C ∞ ( T M ), then we get g ( J X, J Y ) = − g ( X, J Y ) = − pg ( X, J Y ) − qg ( X, Y ) = pg ( J X, Y ) − qg ( X, Y ). On the other hand, g ( J X, J Y ) = − g ( J X, Y ) = − pg ( J X, Y ) − qg ( X, Y ), therefore p = 0 . In particular, for p = 0, it is not possible to define the conceptof metallic Hermitian structure. Definition . [3] i) A linear connection ∇ on M is called J - connection if J iscovariantly constant with respect to ∇ , i.e. ∇ J = 0.ii) A metallic pseudo-Riemannian manifold ( M, J, g ) such that the Levi-Civita connec-tion ∇ with respect to g is a J -connection is called a locally metallic pseudo-Riemannianmanifold . For a metallic pseudo-Riemannian structure (
J, g ) on the smooth manifold M with ∇ theLevi-Civita connection of g , we introduce some tensor fields [7] used to characterize theproperties of the metallic distributions defined by J : oliations induced by metallic structures the J -bracket [ X, Y ] J := [ J X, Y ] + [
X, J Y ] − J ([ X, Y ]) , where [ · , · ] is the Lie bracket, [ X, Y ] = ∇ X Y − ∇ Y X the Nijenhuis tensor associated to JN J ( X, Y ) := J ([ X, Y ] J ) − [ J X, J Y ]3. the Jordan bracket associated to J { X, Y } J := { J X, Y } + { X, J Y } − J ( { X, Y } ) , where {· , ·} is the Jordan bracket, { X, Y } = ∇ X Y + ∇ Y X the Jordan tensor associated to JM J ( X, Y ) := J ( { X, Y } J ) − { J X, J Y } the deformation tensor associated to JH J ( X, Y ) := ( J ◦ ∇ X J − ∇ JX J )( Y )which satisfies 2 H J = N J + M J . Remark . The J -bracket and the associated Nijenhuis tensor can be defined forany (1 , M , the Jordan bracket, the associated Jordantensor and the deformation tensor can be defined for (1 , M, g ).Assume that J satisfies J = pJ + qI with p + 4 q >
0, denote by σ ± := p ± √ p +4 q andconsider the projection operators P and P ′ [8]: P := − p p + 4 q J + σ + p p + 4 q I, P ′ := 1 p p + 4 q J − σ − p p + 4 q I satisfying P = P , P ′ = P ′ , P + P ′ = I, P ◦ P ′ = 0 , P ′ ◦ P = 0 . From a direct computation, we get the following:
Proposition . For the two projection operators P and P ′ :1. N P = N P ′ = p +4 q N J ;2. M P = M P ′ = p +4 q M J ; oliations induced by metallic structures H P = H P ′ = p +4 q H J . Consider now the deformation tensors H and H ′ : H ( X, Y ) := P ′ ( ∇ P X P Y ) = P ′ (( ∇ P X P ) Y ) , H ′ ( X, Y ) := P ( ∇ P ′ X P ′ Y ) = P (( ∇ P ′ X P ′ ) Y ) the twisting tensors L and L ′ : L ( X, Y ) := 12 [ H ( X, Y ) − H ( Y, X )] , L ′ ( X, Y ) := 12 [ H ′ ( X, Y ) − H ′ ( Y, X )]and the extrinsic curvature tensors K and K ′ : K ( X, Y ) := 12 [ H ( X, Y ) + H ( Y, X )] , K ′ ( X, Y ) := 12 [ H ′ ( X, Y ) + H ′ ( Y, X )] , for any X , Y ∈ C ∞ ( T M ).By a direct computation we obtain: H ( X, Y ) = 1( p + 4 q ) p p + 4 q [ J ( ∇ JX J Y ) − σ + J ( ∇ X J Y ) − σ + J ( ∇ JX Y ) + σ J ( ∇ X Y ) −− σ − ∇ JX J Y − q ∇ X J Y − q ∇ JX Y + qσ + ∇ X Y ] == 1( p + 4 q ) p p + 4 q [ J ( ∇ JX J ) − σ + J ( ∇ X J ) − σ − ( ∇ JX J ) − q ( ∇ X J )]( Y ) H ′ ( X, Y ) = − p + 4 q ) p p + 4 q [ J ( ∇ JX J Y ) − σ − J ( ∇ X J Y ) − σ − J ( ∇ JX Y )+ σ − J ( ∇ X Y ) −− σ + ∇ JX J Y − q ∇ X J Y − q ∇ JX Y + qσ − ∇ X Y ] == − p + 4 q ) p p + 4 q [ J ( ∇ JX J ) − σ − J ( ∇ X J ) − σ + ( ∇ JX J ) − q ( ∇ X J )]( Y ) . In particular, we get: H ( X, Y ) + H ′ ( X, Y ) = 1( p + 4 q ) p p + 4 q ( − σ + + σ − )[ J ( ∇ X J ) − ( ∇ JX J )]( Y ) == 1 p + 4 q H J ( X, Y ) . Moreover: L = 12( p + 4 q ) p p + 4 q ( σ − N J − J ◦ N J ) , L ′ = − p + 4 q ) p p + 4 q ( σ + N J − J ◦ N J ) ,K = 12( p + 4 q ) p p + 4 q ( σ − M J − J ◦ M J ) , K ′ = − p + 4 q ) p p + 4 q ( σ + M J − J ◦ M J ) . oliations induced by metallic structures Let (
M, J, g ) be a metallic pseudo-Riemannian manifold such that J = pJ + qI with p + 4 q >
0. Define the complementary distributions:(1) D := ker P ′ , D ′ := ker P which we shall call the metallic distributions defined by the metallic structure J . Remark . The distributions D and D ′ are J -invariant, and, if q = 0, then D and D ′ are also g -orthogonal. Definition . We say that a distribution
D ⊂
T M on a smooth manifold M iscalledi) involutive if X , Y ∈ Γ( D ) implies [ X, Y ] ∈ Γ( D );ii) integrable if for any x ∈ M , there exists a submanifold N x which admits D| N x astangent bundle.According to Frobenius theorem, a distribution D on M is involutive if and only if itis integrable. In this case, it defines a foliation whose leaves are the maximal connectedsubmanifolds N x of M which admit D| N x as tangent bundle. Definition . We say that the metallic pseudo-Riemannian manifold (
M, J, g ) is doubly foliated if both of the distributions D and D ′ given by (1) are integrable and singlyfoliated if only one of them is integrable. Remark . The distribution D (resp. D ′ ) given by (1) is integrable if and only if( ∇ X J ) Y − ( ∇ Y J ) X = 0, for any X , Y ∈ Γ( D ) (resp. X , Y ∈ Γ( D ′ )), with ∇ a torsion-free linear connection on M . Indeed, for X , Y ∈ Γ( D ) we have J X = σ − X , J Y = σ − Y and J ( ∇ X Y − ∇ Y X ) = − ( ∇ X J ) Y + ( ∇ Y J ) X + σ − ( ∇ X Y − ∇ Y X ) which implies that[ X, Y ] ∈ Γ( D ) if and only if ( ∇ X J ) Y − ( ∇ Y J ) X = 0.In particular, in a locally metallic pseudo-Riemannian manifold, the two distributions D and D ′ given by (1) are both integrable. Proposition . If ( M, J, g ) is a metallic pseudo-Riemannian manifold, then thedistribution D is integrable if and only if: J ◦ N J ( X, Y ) = σ − N J ( X, Y ) , for any X, Y ∈ C ∞ ( T M ) , respectively, D ′ is integrable if and only if: J ◦ N J ( X, Y ) = σ + N J ( X, Y ) , for any X, Y ∈ C ∞ ( T M ) . In particular, both D and D ′ are integrable if and only if N J = 0 . oliations induced by metallic structures Proof.
The distribution D is integrable if and only if P ′ ([ P X, P Y ]) = 0 , for any X , Y ∈ C ∞ ( T M ). Therefore, from a direct computation and using Proposition2.5, we obtain that a necessary and sufficient condition for D to be integrable is:0 = P ′ ([ P X, P Y ]) = −P ′ ( N P ( X, Y )) = − p + 4 q P ′ ( N J ( X, Y )) == − p + 4 q ) p p + 4 q [ J ◦ N J ( X, Y ) − σ − N J ( X, Y )] . Definition . Given a linear connection ∇ on a smooth manifold M , we say that adistribution D ⊂
T M is ∇ - geodesically invariant if X , Y ∈ Γ( D ) implies ∇ X Y + ∇ Y X ∈ Γ( D ).In particular, if ∇ is the Levi-Civita of the pseudo-Riemannian manifold ( M, g ), then D is geodesically invariant .Remark that the above condition is equivalent to the following: the distribution D is ∇ -geodesically invariant if X ∈ Γ( D ) implies ∇ X X ∈ Γ( D ). Remark . For a linear connection ∇ on M , the distribution D (resp. D ′ ) givenby (1) is ∇ -geodesically invariant if and only if ( ∇ X J ) Y + ( ∇ Y J ) X = 0, for any X , Y ∈ Γ( D ) (resp. X , Y ∈ Γ( D ′ )). Indeed, for X , Y ∈ Γ( D ) we have J X = σ − X , J Y = σ − Y and J ( ∇ X Y + ∇ Y X ) = − ( ∇ X J ) Y − ( ∇ Y J ) X + σ − ( ∇ X Y + ∇ Y X ) whichimplies that ∇ X Y + ∇ Y X ∈ Γ( D ) if and only if ( ∇ X J ) Y + ( ∇ Y J ) X = 0.In particular, for any J -connection ∇ , the distributions D and D ′ are ∇ -geodesicallyinvariant. Proposition . If ( M, J, g ) is a metallic pseudo-Riemannian manifold, then thedistribution D is geodesically invariant if and only if: J ◦ M J ( X, Y ) = σ − M J ( X, Y ) , for any X, Y ∈ C ∞ ( T M ) , respectively, D ′ is geodesically invariant if and only if: J ◦ M J ( X, Y ) = σ + M J ( X, Y ) , for any X, Y ∈ C ∞ ( T M ) . In particular, both D and D ′ are geodesically invariant if and only if M J = 0 . oliations induced by metallic structures Proof.
The distribution D is geodesically invariant if and only if P ′ ( {P X, P Y } ) = 0 , for any X , Y ∈ C ∞ ( T M ). Therefore, from a direct computation and using Proposition2.5, we obtain that a necessary and sufficient condition for D to be geodesically invariantis: 0 = P ′ ( {P X, P Y } ) = −P ′ ( M P ( X, Y )) = − p + 4 q P ′ ( M J ( X, Y )) == − p + 4 q ) p p + 4 q [ J ◦ M J ( X, Y ) − σ − M J ( X, Y )] . Remark . J p := P − P ′ is an almost product structure on M and J p X = − p p + 4 q (2 J − pI ) X, for any X ∈ C ∞ ( T M ).Direct computations provide the following relationship between J and J p -brackets, J and J p Nijenhuis tensors, Jordan bracket and Jordan tensors of the two structures.Precisely, we have the following:
Proposition . [ X, Y ] J = − p p + 4 q X, Y ] J p + p X, Y ] N J ( X, Y ) = p + 4 q N J p ( X, Y ) { X, Y } J = − p p + 4 q { X, Y } J p + p { X, Y } M J ( X, Y ) = p + 4 q M J p ( X, Y ) . In particular, the deformation tensors are related as follows: H J ( X, Y ) = p + 4 q H J p ( X, Y ) . The product conjugate connection of a linear connection ∇ is [2]:(2) ∇ ( J p ) X Y = P ( ∇ X P Y ) − P ( ∇ X P ′ Y ) − P ′ ( ∇ X P Y ) + P ′ ( ∇ X P ′ Y )and we have: oliations induced by metallic structures Proposition . [2] If ∇ ( J p ) is torsion-free, then J p is integrable, which means that D and D ′ are integrable distributions. Definition . We say that a linear connection ∇ restricts to a distribution D ⊂
T M on a metallic pseudo-Riemannian manifold (
M, J, g ) if Y ∈ Γ( D ) implies ∇ X Y ∈ Γ( D ), for any X ∈ C ∞ ( T M ).We have:1) ∇ restricts to D means P ′ ( ∇ X P Y ) = 0 and P ( ∇ X P Y ) = ∇ X P Y ,2) ∇ restricts to D ′ means P ( ∇ X P ′ Y ) = 0 and P ′ ( ∇ X P ′ Y ) = ∇ X P ′ Y .A straightforward computation gives that the product conjugate connection ∇ ( J p ) defined by (2) restricts to D and D ′ . Moreover, if ∇ restricts to both D and D ′ , then(3) ∇ ( J p ) X Y = ∇ X P Y + ∇ X P ′ Y = ∇ X Y and so ∇ is an J p -connection. Let us remark that the above connection (3) is exactly theSchouten-van Kampen connection of the pair ( D , D ′ ): ∇ X Y = P ( ∇ X P Y ) + P ′ ( ∇ X P ′ Y )which coincides with the metallic natural connection ˜ ∇ if ∇ is the Levi-Civita connectionof g .Now we can express the Kirichenko tensor fields [9] in terms of the projectors P , P ′ : Proposition . [2] The structural and virtual tensor fields of J p = P − P ′ are: ( C P−P ′ ∇ ( X, Y ) = 2[ P ( ∇ P ′ X P ′ Y ) + P ′ ( ∇ P X P Y )] B P−P ′ ∇ ( X, Y ) = − P ( ∇ P X P ′ Y ) + P ′ ( ∇ P ′ X P Y )] . Let us recall the well-known fundamental tensor fields of O’Neill-Gray: ( T ( X, Y ) = P ( ∇ P ′ X P ′ Y ) + P ′ ( ∇ P ′ X P Y ) A ( X, Y ) = P ′ ( ∇ P X P Y ) + P ( ∇ P X P ′ Y ) . Then, a comparison of last two equations yields ( C P−P ′ ∇ ( X, Y ) = 2[ T ( X, P ′ Y ) + A ( X, P Y )] B P−P ′ ∇ ( X, Y ) = − T ( X, P Y ) + A ( X, P ′ Y )]a fact which justifies the second name of T and A as invariants of the decomposition T M = D ⊕ D ′ [6].On D with the induced metric g D , we consider the induced connection from the pseudo-Riemannian manifold ( M, g, ∇ ) by [10]: ∇ D : Γ( D ) × Γ( D ) → Γ( D ) , ∇ D X Y := P ( ∇ X Y ) oliations induced by metallic structures g D and torsion-free w.r.t. the bracket[ · , · ] D : Γ( D ) × Γ( D ) → Γ( D ) , [ X, Y ] D := P ([ X, Y ]) . The bracket [ · , · ] D has the usual properties of a Lie bracket excepting Jacobi identity whichis satisfied if and only if D is integrable.The integrability of D can also be characterized in terms of second fundamental formof D : h : Γ( D ) × Γ( D ) → Γ( D ′ ) , h ( X, Y ) := ∇ X Y − ∇ D X Y, and we can state: Proposition . [10] The distribution D is integrable if and only if one of thefollowing assertions holds: i) ∇ D is torsion-free; ii) h is symmetric. Similarly, on ( D ′ , g D ′ ) we define the induced connection from ( M, g, ∇ ) by: ∇ D ′ : Γ( D ′ ) × Γ( D ′ ) → Γ( D ′ ) , ∇ D ′ X Y := P ′ ( ∇ X Y )and consider the second fundamental form h ′ of D ′ . Then the distribution D ′ is integrableif and only if one of the following assertions holds: i) ∇ D ′ is torsion-free; ii) h ′ is symmetric.Remark that the restrictions of the metallic natural connection ˜ ∇ , defined in [3], to D and respectively, to D ′ , coincide with the two induced connections, respectively:˜ ∇| Γ( D ) × Γ( D ) = ∇ D , ˜ ∇| Γ( D ′ ) × Γ( D ′ ) = ∇ D ′ . Remark . For p + 4 q = 0, we get only one distribution, ker( J − p I ), and J t := J − p I is an almost subtangent structure. ( D , D ′ ) Definition . We say that a linear connection ∇ on M is adapted to the decom-position T M = D ⊕ D ′ if Y ∈ Γ( D ) implies ∇ X Y ∈ Γ( D ), for any X ∈ C ∞ ( T M ) and Y ∈ Γ( D ′ ) implies ∇ X Y ∈ Γ( D ′ ), for any X ∈ C ∞ ( T M ). Remark . If (
M, J ) is a metallic manifold such that J = pJ + qI with p + 4 q > ∇ is adapted to ( D , D ′ ) given by (1) if and only if ∇ is a J -connection. Indeed, for Y ∈ Γ( D ) we have J Y = σ − Y and ( ∇ X J ) Y = σ − ∇ X Y − J ( ∇ X Y ),for any X ∈ C ∞ ( T M ), which implies that ∇ X Y ∈ Γ( D ) if and only if ∇ J = 0. Similarlywe deduce the second implication. oliations induced by metallic structures D , D ′ ), namely:(4) ∇ ∗ X Y = P ( ∇ X P Y ) + P ′ ( ∇ X P ′ Y ) + P ( S ( X, P Y )) + P ′ ( S ( X, P ′ Y )) , for any X , Y ∈ C ∞ ( T M ), where ∇ is a linear connection and S is a (1 , M . An adapted connection to ( D , D ′ ) is the Schouten-van Kampen connection ˜ ∇ of the linearconnection ∇ , obtained from (4) for S := 0:(5) ˜ ∇ X Y := P ( ∇ X P Y ) + P ′ ( ∇ X P ′ Y ) == ∇ X Y + P (( ∇ X P ) Y ) + P ′ (( ∇ X P ′ ) Y ) . If (
M, J, g ) is a metallic pseudo-Riemannian manifold such that J = pJ + qI with p + 4 q > ∇ is torsion-free, then ˜ ∇ is explicitly given by:(6) ˜ ∇ X Y = 1 p + 4 q [(2 J − pI )( ∇ X J Y ) − ( pJ − ( p + 2 q ) I )( ∇ X Y )] , for any X , Y ∈ C ∞ ( T M ). Remark that if ∇ is the Levi-Civita connection associatedto g , then ˜ ∇ is exactly the metallic natural connection defined in [3]. Moreover, ˜ ∇ is ametric J -connection, i.e. ˜ ∇ g = ˜ ∇ J = 0, whose torsion is given by: T ˜ ∇ ( X, Y ) = 1 p + 4 q [(2 J − pI )( ∇ X J Y − ∇ Y J X ) − ( pJ + 2 qI )( ∇ X Y − ∇ Y X )] , for any X , Y ∈ C ∞ ( T M ). Another adapted connection to ( D , D ′ ) is Vr˘anceanu connection ¯ ∇ of the linear connection ∇ , obtained from (4) for S ( X, Y ) := −P ( ∇ P ′ X P Y ) − P ′ ( ∇ P ′ X P ′ Y ) + P ([ P ′ X, P Y ]) + P ′ ([ P X, P ′ Y ]) . If (
M, J, g ) is a metallic pseudo-Riemannian manifold such that J = pJ + qI with p + 4 q >
0, then ¯ ∇ is explicitly given by:(7) ¯ ∇ X Y = ˜ ∇ P X Y + P ([ P ′ X, P Y ]) + P ′ ([ P X, P ′ Y ]) == ∇ X Y + 1 p + 4 q [2 J (( ∇ X J ) Y ) − p ( ∇ X J ) Y + J (( ∇ Y J ) X ) + ( ∇ JY J ) X − p ( ∇ Y J ) X ]+ oliations induced by metallic structures
11+ 1 p + 4 q [ T ∇ ( J X, J Y ) + J ( T ∇ ( J X, Y )) − pT ∇ ( J X, Y ) − J ( T ∇ ( X, J Y )) − qT ∇ ( X, Y )] , for any X , Y ∈ C ∞ ( T M ).Moreover, ¯ ∇ is a J -connection, i.e. ¯ ∇ J = 0, whose torsion is given by: T ¯ ∇ ( X, Y ) = 1 p + 4 q N J ( X, Y ) + P ′ ( T ∇ ( P ′ X, P ′ Y )) − P ( T ∇ ( P X, P Y )) , for any X , Y ∈ C ∞ ( T M ). Let (
M, J, g ) be a metallic pseudo-Riemannian manifold such that J = pJ + qI with p + 4 q > ∇ be the Levi-Civita connection of g .Another adapted connection to ( D , D ′ ) is the Vidal connection ˜˜ ∇ associated to J ,obtained from (4) for S ( X, Y ) := −P ( ∇ P Y P ′ ) X − P ′ ( ∇ P ′ Y P ) X, therefore:(8) ˜˜ ∇ X Y = ˜ ∇ X Y − P ( ∇ P Y P ′ ) X − P ′ ( ∇ P ′ Y P ) X == ˜ ∇ X Y + 1 p + 4 q [( ∇ JY J ) X + J (( ∇ Y J ) X ) − p ( ∇ Y J ) X ] == ∇ X Y + 1 p + 4 q [2 J (( ∇ X J ) Y ) − p ( ∇ X J ) Y + J (( ∇ Y J ) X ) + ( ∇ JY J ) X − p ( ∇ Y J ) X ] , for any X , Y ∈ C ∞ ( T M ).Moreover, ˜˜ ∇ is a J -connection, i.e. ˜˜ ∇ J = 0, whose torsion is given by: T ˜˜ ∇ ( X, Y ) = 1 p + 4 q N J ( X, Y ) , for any X , Y ∈ C ∞ ( T M ). Remark . Vr˘anceanu connection of the Levi-Civita connection coincides with theVidal connection.Moreover, we get:( ˜˜ ∇ X g )( Y, Z ) = − p + 4 q [ g (( ∇ JY J ) X − ( ∇ Y J ) J X, Z ) + g (( ∇ JZ J ) X − ( ∇ Z J ) J X, Y )] == 1 p + 4 q [ g ( M J ( Y, X ) , Z ) + g ( M J ( Z, X ) , Y )++ g (( ∇ JX J ) Y + ( ∇ Y J ) J X, Z ) + g (( ∇ JX J ) Z + ( ∇ Z J ) J X, Y )] , for any X , Y , Z ∈ C ∞ ( T M ).Since ˜ ∇ J = ¯ ∇ J = ˜˜ ∇ J = 0, from Remark 3.7 we deduce: oliations induced by metallic structures Proposition . The distributions D and D ′ are ˜ ∇ -geodesically invariant, ¯ ∇ -geo-desically invariant and ˜˜ ∇ -geodesically invariant. Using the Vidal connection ˜˜ ∇ , we characterize the integrability and the geodesicallyinvariance of the metallic distributions defined by J in terms of the torsion and thecovariant derivative of g w.r.t. to this connection. From all the above considerations, wecan state: Theorem . If ( M, J, g ) is a metallic pseudo-Riemannian manifold such that J = pJ + qI with p + 4 q > , then the following assertions are equivalent:(i) the distributions D and D ′ are integrable;(ii) N J = 0 ;(iii) L = 0 and L ′ = 0 ;(iv) the Vidal connection given by (8) is torsion-free. Theorem . If ( M, J, g ) is a metallic pseudo-Riemannian manifold such that J = pJ + qI with p + 4 q > , then the following assertions are equivalent:(i) the distributions D and D ′ are geodesically invariant;(ii) M J = 0 ;(iii) K = 0 and K ′ = 0 ;(iv) the Vidal connection given by (8) is metric with respect to g . We shall provide the condition for a metallic map between two metallic pseudo-Riemannianmanifolds to preserve the metallic distributions. We recall the following:
Definition . A smooth map Φ : ( M , J ) → ( M , J ) between two metallic mani-folds is called a metallic map if: Φ ∗ ◦ J = J ◦ Φ ∗ . Remark . If Φ : ( M , J ) → ( M , J ) is a metallic map and J i = p i J i + q i I with p i and q i real numbers, i = 1 ,
2, then:i) Φ ∗ ◦ J k +11 = J k +12 ◦ Φ ∗ , for any k ∈ N ;ii) ([( p + q ) − ( p + q )] J + ( p q − p q ) I )( T M ) ⊂ ker Φ ∗ ;iii) in the particular case when one the structure is product and the other one iscomplex, then ImJ ⊂ ker Φ ∗ .Consider a metallic map Φ : ( M , J ) → ( M , J ) between the metallic manifolds( M i , J i ) such that J i = p i J i + q i I with p i + 4 q i > i = 1 ,
2, and assume that the oliations induced by metallic structures D i and D ′ i , i = 1 ,
2, are integrable. Then they define the foliations F i and F ′ i , i = 1 ,
2, whose leaves are trivial metallic pseudo-Riemannian manifolds.Denoting by Φ ∗ D the pull-back distribution, i.e.:(Φ ∗ D ) x := { X x ∈ T x M : Φ ∗ x ( X x ) ∈ D x ) } , since Φ is a metallic map, we get:(Φ ∗ D ) x = { X x ∈ T x M : ( J − σ I )( X x ) ∈ ker Φ ∗ x } , where σ i + = p i + √ p i +4 q i , i = 1 , ∗ D ′ ) x = { X x ∈ T x M : ( J − σ − I )( X x ) ∈ ker Φ ∗ x } , where σ i − = p i − √ p i +4 q i , i = 1 , ∗ D to coincide with one of the distributions D or D ′ : Proposition . If ker Φ ∗ = ( J − σ I )(ker( J − σ I )) , then Φ ∗ D = D . More-over, if Φ is a surjective submersion with connected fibers, then a leaf of F correspondsto a leaf of F . A fundamental problem in the theory of submanifolds is the problem posed by B. Y.Chen [4], namely, to find relations between the main intrinsic and extrinsic invariants of asubmanifold. In this sense, the Chen’s inequalities for submanifolds in real space forms wasproved by B. Y. Chen [4], in complex space forms by Y. Do˘gru [5], in quaternionic spaceforms by G. E. Vˆılcu [12] etc. In the same spirit, we shall prove a Chen-type inequalityin the metallic case, for an integrable distribution defined by the metallic structure.Let (
M, J, g ) be an m -dimensional metallic Riemannian manifold and assume that thedistribution D is integrable. In this case, the Riemann curvature tensors of D (computedwith respect to the induced connection ∇ D on D and the Lie bracket [ · , · ] D ) and M satisfy[10]:(9) R D ( X, Y, Z, W ) = R M ( X, Y, Z, W ) − g ( h ( X, Z ) , h ( Y, W )) + g ( h ( X, W ) , h ( Y, Z )) , for any X , Y , Z , W ∈ Γ( D ). oliations induced by metallic structures J -sectional curva-ture, is given in the followings.From a direct computation we obtain: Proposition . Let ( M, J, g ) be an m -dimensional metallic Riemannian manifold( m > ) such that J = pJ + qI with p + 4 q > , whose Riemann curvature tensor isgiven by (10) R M ( X, Y, Z, W ) = c [ g ( X, F W ) g ( Y, F Z ) − g ( X, F Z ) g ( Y, F W )] , for any X , Y , Z , W ∈ C ∞ ( T M ) , where F := aJ + bI with a and b real numbers satisfying qa − pab − b = 1 . Then the J -sectional curvature of M is constant equal to c . Denote by H := n tr ( h ) the mean curvature and by δ D := τ D − inf K D the Chenfirst invariant of D , where τ D denotes the scalar curvature of D and K D its sectionalcurvature. Theorem . Let ( M, J, g ) be an m -dimensional metallic Riemannian manifold ( m > ) such that J = pJ + qI with p + 4 q > , whose Riemann curvature tensor is given by(10) and let D given by (1) be an n -dimensional integrable distribution. Then: δ D ≤ c ( aσ − + b ) ( n − n + 2)2 + n ( n − n − || H || . Proof.
Consider an orthonormal frame field { e , . . . , e n } for D , { f , . . . , f m − n } anorthonormal frame field for D ′ and denote by h kij := g ( h ( e i , e j ) , f k ) . From (9) and (10) we get2 τ D = c ( aσ − + b ) n ( n − − || h || + n || H || . Moreover K D ( e , e ) = − c ( aσ − + b ) − m − n X k =1 h k h k + m − n X k =1 ( h k ) and τ D − K D ( e , e ) = c ( aσ − + b ) ( n − n + 2)2 ++ m − n X k =1 [ X ≤ i M, J, g ) be a metallic Norden manifold such that J = pJ + qI with p + 4 q < T C M := T M ⊗ R C be the complexified tangent bundle. Then we can define thecomplexified metallic pseudo-Riemannian structure : J C ( X + iY ) := J X + iJ Y,g C ( X + iY , X + iY ) := g ( X , X ) − g ( Y , Y ) + i [ g ( X , Y ) + g ( Y , X )] , for any X , X , X , Y , Y , Y ∈ C ∞ ( T M ).Denote by σ C ± := p ± √ p +4 q and consider the projection operators P C and P C ′ : P C := − p p + 4 q J C + σ C + p p + 4 q I C , P C ′ := 1 p p + 4 q J C − σ C − p p + 4 q I C satisfying P C = P C , P C ′ = P C ′ , P C + P C ′ = I C , P C ◦ P C ′ = 0 , P C ′ ◦ P C = 0and define the complementary distributions:(11) D C := ker P C ′ , D C ′ := ker P C which we shall call the complex metallic distributions defined by J . oliations induced by metallic structures Remark . If ( M, J, g ) is a metallic Norden manifold such that J = pJ + qI with p + 4 q < 0, then D C and D C ′ are J C -invariant, and, if q = 0, then D C and D C ′ are also g C -orthogonal. Lemma . D C ′ = D C Proof. It follows from the following: σ C + = p + p p + 4 q p + i p − p − q p − i p − p − q p − p p + 4 q σ C − . In particular, if J is not trivial, that it admits two complex eigenvalues, or the twodistributions are both different from 0, then the complexified tangent bundle splits asdirect sum of two conjugate subbundles: T C M = D C ⊕ D C . Extending the Lie bracket to:[ X + iY , X + iY ] C := [ X , X ] − [ Y , Y ] + i ([ X , Y ] + [ Y , X ]) , for any X , X , Y , Y ∈ C ∞ ( T M ), we say that: Definition . A distribution D C ⊂ T C M is called integrable if X , Y ∈ Γ( D C )implies [ X, Y ] C ∈ Γ( D C ). Lemma . The distribution D C is integrable if and only if P C ′ ([ P C X, P C Y ] C ) = 0 , for any X , Y ∈ C ∞ ( T C M ) . Proposition . The distribution D C (resp. D C ′ ) given by (11) is integrable if andonly if N J = 0 . Extending the Levi-Civita connection ∇ of g to: ∇ C X + iY ( X + iY ) := ∇ X X − ∇ Y Y + i ( ∇ X Y + ∇ Y X ) , for any X , X , Y , Y ∈ C ∞ ( T M ), we pose the following: oliations induced by metallic structures Definition . Given a complex linear connection ∇ C on a smooth manifold M ,a distribution D C ⊂ T C M is called ∇ C - geodesically invariant if X , Y ∈ Γ( D C ) implies ∇ C X Y + ∇ C Y X ∈ Γ( D C ).In particular, if ∇ C is the Levi-Civita connection of the pseudo-Riemannian manifold( M, g C ), then D C is called geodesically invariant . Lemma . The distribution D C is geodesically invariant if and only if P C ′ ( {P C X, P C Y } C ) = 0 , for any X , Y ∈ C ∞ ( T C M ) , where { X, Y } C := ∇ C X Y + ∇ C X Y . Proposition . The distribution D C (resp. D C ′ ) given by (11) is geodesically in-variant if and only if M J = 0 . Remark . For a complex linear connection ∇ C on M , the distribution D C (resp. D C ′ ) given by (11) is ∇ C -geodesically invariant if and only if ( ∇ C X J C ) Y + ( ∇ C Y J C ) X = 0,for any X , Y ∈ Γ( D C ) (resp. X , Y ∈ Γ( D C ′ )). Indeed, for X , Y ∈ Γ( D C ) we have J C X = σ C − X , J C Y = σ C − Y and J C ( ∇ C X Y + ∇ C Y X ) = − ( ∇ C X J C ) Y − ( ∇ C Y J C ) X + σ C − ( ∇ C X Y + ∇ C Y X )which implies that ∇ C X Y + ∇ C Y X ∈ Γ( D C ) if and only if ( ∇ C X J C ) Y + ( ∇ C Y J C ) X = 0.In particular, for any J C -connection ∇ C , the distributions D C and D C ′ are ∇ C -geode-sically invariant. Remark . J c := i ( P C − P C ′ ) is a Norden structure on M and J c X = − p − p − q (2 J − pI ) X, for any X ∈ C ∞ ( T M ).By a direct computation we get: Proposition . The Nijenhuis tensors of J c and J are related as follows: N J c ( X, Y ) = 4 − p − q N J ( X, Y ) , for any X, Y ∈ C ∞ ( T M ) . Moreover, if T C M = T (1 , M ⊕ T (0 , M is the decomposition of the complexified tangent bundle into (1 , 0) and (0 , 1) parts, withrespect to the almost complex structure J c , we have: D C ′ = T (1 , M and D C = T (0 , M. oliations induced by metallic structures Definition . We say that a complex linear connection ∇ C on M is adapted tothe decomposition T C M = D C ⊕ D C ′ if Y ∈ Γ( D C ) implies ∇ C X Y ∈ Γ( D C ), for any X ∈ C ∞ ( T C M ) and Y ∈ Γ( D C ′ ) implies ∇ X Y ∈ Γ( D C ′ ), for any X ∈ C ∞ ( T C M ). Remark . If ( M, J, g ) is a metallic Norden manifold such that J = pJ + qI with p + 4 q < 0, then a complex linear connection ∇ C is adapted to ( D C , D C ′ ) given by(11) if and only if ∇ C is a J C -connection. Indeed, for Y ∈ Γ( D C ) we have J C Y = σ C − Y and ( ∇ C X J C ) Y = σ C − ∇ C X Y − J C ( ∇ C X Y ), for any X ∈ C ∞ ( T C M ), which implies that ∇ C X Y ∈ Γ( D C ) if and only if ∇ C J C = 0. Similarly we deduce the second implication. Proposition . All adapted connections to ( D C , D C ′ ) are of the form: (12) ( ∇ C ) ∗ X Y = P C ( ∇ C X P C Y ) + P C ′ ( ∇ C X P C ′ Y ) + P C ( S ( X, P C Y )) + P C ′ ( S ( X, P C ′ Y )) , for any X , Y ∈ C ∞ ( T C M ) , where ∇ C is a complex linear connection and S is a complex (1 , -tensor field on M . Proof. We follow the same steps like in the real case [1].Consider the following adapted connection to ( D C , D C ′ ):1) The complex Schouten-van Kampen connection ˜ ∇ C of the complex linear connection ∇ C , obtained from (12) for S := 0:˜ ∇ C X Y := P C ( ∇ C X P C Y ) + P C ′ ( ∇ C X P C ′ Y ) . If ( M, J, g ) is a metallic Norden manifold such that J = pJ + qI with p + 4 q < ∇ C is torsion-free, then ˜ ∇ C is explicitly given by:(13) ˜ ∇ C X Y = 1 p + 4 q [(2 J C − pI C )( ∇ C X J Y ) − ( pJ C − ( p + 2 q ) I C )( ∇ C X Y )] == ∇ C X Y + 1 p + 4 q [2 J C ( ∇ C X J C ) − p ( ∇ C X J C )] Y, for any X , Y ∈ C ∞ ( T C M ).Remark that if ∇ C is the Levi-Civita connection associated to g C , then ˜ ∇ C is a metric J C -connection, i.e. ˜ ∇ C g C = ˜ ∇ C J C = 0, whose torsion is given by: T ˜ ∇ C ( X, Y ) = 1 p + 4 q [(2 J C − pI C )( ∇ C X J Y − ∇ C Y J C X ) − ( pJ C + 2 qI C )( ∇ C X Y − ∇ C Y X )] , for any X , Y ∈ C ∞ ( T C M ). oliations induced by metallic structures The complex Vr˘anceanu connection ¯ ∇ C of the complex linear connection ∇ C , ob-tained from (12) for S ( X, Y ) := −P C ( ∇ C P C ′ X P C Y ) −P C ′ ( ∇ C P C ′ X P C ′ Y )+ P C ([ P C ′ X, P C Y ] C )+ P C ′ ([ P C X, P C ′ Y ] C ) . If ( M, J, g ) is a metallic Norden manifold such that J = pJ + qI with p + 4 q < ∇ C is explicitly given by:(14) ¯ ∇ C X Y = ˜ ∇ C P C X Y + P C ([ P C ′ X, P C Y ] C ) + P C ′ ([ P C X, P C ′ Y ] C ) , for any X, Y ∈ C ∞ ( T C M ).Moreover, ¯ ∇ C is a J C -connection, i.e. ¯ ∇ C J C = 0, whose torsion is given by: T ¯ ∇ C ( X, Y ) = 1 p + 4 q N J C ( X, Y ) + P C ′ ( T ∇ C ( P C ′ X, P C ′ Y )) − P C ( T ∇ C ( P C X, P C Y )) , for any X , Y ∈ C ∞ ( T C M ).3) The complex Vidal connection ˜˜ ∇ C associated to the metallic Norden structure ( J, g ),obtained from (12) for S ( X, Y ) := −P C ( ∇ P C Y P C ′ ) X − P C ′ ( ∇ P C ′ Y P C ) X, therefore:(15) ˜˜ ∇ C X Y = ˜ ∇ C X Y − P C ( ∇ P C Y P C ′ ) X − P C ′ ( ∇ P C ′ Y P C ) X == ˜ ∇ C X Y + 1 p + 4 q [( ∇ J C Y J C ) X + J C (( ∇ Y J C ) X ) − p ( ∇ Y J C ) X ] , for any X, Y ∈ C ∞ ( T C M ), where ∇ C is the Levi-Civita connection of g C .Moreover, ˜˜ ∇ C is a J C -connection, i.e. ˜˜ ∇ C J C = 0, whose torsion is given by: T ˜˜ ∇ C ( X, Y ) = 1 p + 4 q N J C ( X, Y ) , for any X, Y ∈ C ∞ ( T C M ).Moreover, we get:( ˜˜ ∇ C X g C )( Y, Z ) = − p + 4 q [ g C (( ∇ C J C Y J C ) X − ( ∇ C Y J C ) J C X, Z )++ g C (( ∇ J C Z J C ) X − ( ∇ C Z J C ) J C X, Y )] == 1 p + 4 q [ g C ( M J C ( Y, X ) , Z ) + g C ( M J C ( Z, X ) , Y )++ g C (( ∇ C J C X J C ) Y + ( ∇ C Y J C ) J C X, Z ) + g C (( ∇ C J C X J C ) Z + ( ∇ C Z J ) J C X, Y )] , for any X , Y , Z ∈ C ∞ ( T C M ).Since ˜ ∇ C J C = ¯ ∇ C J C = ˜˜ ∇ C J C = 0, from Remark 6.9 we deduce: oliations induced by metallic structures Proposition . The distributions D C and D C ′ are ˜ ∇ C -geodesically invariant, ¯ ∇ C -geodesically invariant and ˜˜ ∇ C -geodesically invariant. From all the above considerations, we can state: Theorem . If ( M, J, g ) is a metallic Norden manifold such that J = pJ + qI with p + 4 q < , then the following assertions are equivalent:(i) the distributions D C and D C ′ are integrable;(ii) ( M, J c ) is a complex manifold;(iii) the complex Vidal connection given by (15) is torsion-free. Theorem . If ( M, J, g ) is a metallic Norden manifold such that J = pJ + qI with p + 4 q < , then the following assertions are equivalent:(i) the distributions D C and D C ′ are geodesically invariant;(ii) the complex Vidal connection given by (15) is metric with respect to g C . ¯ ∂ -operator of a metallic complex structure Definition . A metallic manifold ( M, J ) such that J = pJ + qI with p + 4 q < J integrable is called metallic complex manifold .Let ( M, J ) be a metallic complex manifold and let J c = − √ − p − q (2 J − pI ) be theassociated complex structure. Consider its dual map J ∗ c : T ∗ M → T ∗ M , defined by( J ∗ c α )( X ) := α ( J c X ), for any α ∈ C ∞ ( T ∗ M ) and for any X ∈ C ∞ ( T M ).We shall define the real differential operator d c acting on forms: d c := J ∗ c ◦ d ◦ J ∗ c , where d is the real differential operator.If ( M, J, g ) is an integrable metallic Norden manifold, we can consider the real codif-ferential operator δ c acting on forms: δ c := ⋆ ◦ d c ◦ ⋆, where ⋆ is the Hodge-star operator with respect to the metric g .We obtain d c ◦ d c = 0 , d ◦ d c + d c ◦ d = 0 ,δ c ◦ δ c = 0 , δ ◦ δ c + δ c ◦ δ = 0 , where δ is the codifferential operator, and with respect to the scalar product h· , ·i inducedby g , the operators d c and δ c are adjoint, i.e. h d c α, β i = h α, δ c β i , oliations induced by metallic structures α , β ∈ C ∞ ( T ∗ M ).Remark that J ∗ ◦ ⋆ = ⋆ ◦ J ∗ (and J ∗ c ◦ ⋆ = ⋆ ◦ J ∗ c ) implies δ c = J ∗ c ◦ δ ◦ J ∗ c and d c ◦ J ∗ c = − J ∗ c ◦ d, J ∗ c ◦ d c = − d ◦ J ∗ c ,δ c ◦ J ∗ c = − J ∗ c ◦ δ, J ∗ c ◦ δ c = − δ ◦ J ∗ c . From the above relations, we can state: Proposition . Let α be a real form on M .i) If α is d c -closed (resp. δ c -coclosed), then J ∗ c α is closed (resp. coclosed).ii) If α is closed (resp. coclosed), then J ∗ c α is d c -closed (resp. δ c -coclosed).iii) If α is J ∗ c -invariant, i.e. J ∗ c α = α , then α is d c -closed (resp. δ c -coclosed) if andonly if it is closed (resp. coclosed). Therefore, the d c -closed (resp. δ c -coclosed) forms are the J ∗ c -invariant closed (resp.coclosed) forms. Thenker( d c ) = ker( d ) ∩ { J ∗ c − invariant forms } , Im ( d c ) = J ∗ c ( Im ( d )) , ker( δ c ) = ker( δ ) ∩ { J ∗ c − invariant forms } , Im ( δ c ) = J ∗ c ( Im ( δ )) . Then we can consider the metallic cohomology groups H r ( M ) := ker( d cr ) /Im ( d cr − ) , where d cr : C ∞ (Λ r ( M )) → C ∞ (Λ r +1 ( M ))and the metallic homology groups H r ( M ) := ker( δ cr ) /Im ( δ cr +1 ) , where δ cr : C ∞ (Λ r ( M )) → C ∞ (Λ r − ( M )) . Now we can introduce the metallic Hodge-Laplace operator ∆ c : C ∞ (Λ r ( M )) → C ∞ (Λ r ( M )) , ∆ c := d c ◦ δ c + δ c ◦ d c , which is symmetric and self-adjoint w.r.t. h· , ·i . Remark that∆ c = − J ∗ c ◦ ∆ ◦ J ∗ c , where ∆ = d ◦ δ + δ ◦ d is the Hodge-Laplace operator, and ∆ c satisfies∆ c ◦ J ∗ c = J ∗ c ◦ ∆ , J ∗ c ◦ ∆ c = ∆ ◦ J ∗ c . oliations induced by metallic structures Definition . A real form α is called J -harmonic if it belongs to the kernel of themetallic Hodge-Laplace operator, i.e. ∆ c α = 0.From the above relations, we get: Proposition . Let α be a real form on M .i) If α is J -harmonic, then J ∗ c α is harmonic.ii) If α is harmonic, then J ∗ c α is J -harmonic.iii) If α is J ∗ c -invariant, i.e. J ∗ c α = α , then α is J -harmonic if and only if it isharmonic.iv) α is J -harmonic if and only if it is d c -closed and δ c -coclosed. Therefore, the J -harmonic forms are the J ∗ c -invariant harmonic forms. Thenker(∆ c ) = ker(∆) ∩ { J ∗ c − invariant forms } , Im (∆ c ) = J ∗ c ( Im (∆)) . Let T C M = T (1 , M ⊕ T (0 , M = D C ′ ⊕ D C be the decomposition of the complexified tangent bundle into (1 , 0) and (0 , 1) parts, withrespect to the complex structure J c or, equivalently, with respect to the distributionsdefined by J .The ¯ ∂ -operator and ¯¯ ∂ -operator acting on ( r, s )-forms on M are defined as follows:¯ ∂ : C ∞ (Λ ( r,s ) ( M )) → C ∞ (Λ ( r,s +1) ( M )) , ¯ ∂ := 12 ( d − id c ) , ¯¯ ∂ : C ∞ (Λ ( r,s +1) ( M )) → C ∞ (Λ ( r,s ) ( M )) , ¯¯ ∂ := 12 ( δ − iδ c ) . Remark that the integrability of J (which is equivalent to the integrability of J c )implies ¯ ∂ ◦ ¯ ∂ = 0 , ¯¯ ∂ ◦ ¯¯ ∂ = 0 , therefore we can consider the metallic complex cohomology groups H ( r,s ) ( M ) := ker( ¯ ∂ ( r,s ) ) /Im ( ¯ ∂ ( r,s − ) , where ¯ ∂ ( r,s ) : C ∞ (Λ ( r,s ) ( M )) → C ∞ (Λ ( r,s +1) ( M ))and the metallic complex homology groups H ( r,s ) ( M ) := ker( ¯¯ ∂ ( r,s ) ) /Im ( ¯¯ ∂ ( r,s +1) ) , oliations induced by metallic structures ∂ ( r,s ) : C ∞ (Λ ( r,s ) ( M )) → C ∞ (Λ ( r,s − ( M )) . Now, if T ∗ C M = D ∗ C ⊕ D ∗ C is the decomposition of the complexified cotangent bundle defined by J ∗ , then we get thefollowing: Proposition . Let ( M, J ) be a metallic complex manifold such that J = pJ + qI with p + 4 q < . Then the ¯ ∂ -operator: ¯ ∂ = 12( p + 4 q ) [( p + 4 q ) d + i (4 J ∗ ◦ d ◦ J ∗ − pd ◦ J ∗ − pJ ∗ ◦ d + p d )] is acting on C ∞ (Λ r ( D ∗ )) ⊗ C ∞ (Λ s ( D ∗ C )) . Proof. We have: d c = [ − p − p − q (2 J ∗ − pI )] ◦ d ◦ [ − p − p − q (2 J ∗ − pI )] == − p + 4 q (4 J ∗ ◦ d ◦ J ∗ − pd ◦ J ∗ − pJ ∗ ◦ d + p d ) . Then the statement.Similarly, we prove that: Proposition . Let ( M, J, g ) be a metallic Norden manifold such that J = pJ + qI with p + 4 q < . Then the ¯¯ ∂ -operator: ¯¯ ∂ = 12( p + 4 q ) [( p + 4 q ) δ + i (4 J ∗ ◦ δ ◦ J ∗ − pδ ◦ J ∗ − pJ ∗ ◦ δ + p δ )] is acting on C ∞ (Λ r ( D ∗ )) ⊗ C ∞ (Λ s ( D ∗ C )) . Remark . The operators d c and ¯ ∂ can be defined on metallic complex manifoldsand δ c , ∆ c and ¯¯ ∂ only on metallic Norden manifolds. References [1] A. Bejancu, H. R. Farran, Foliations and geometric structures , Mathematics and Its Appli-cations , Springer, Dordrecht, 2006.[2] A. M. Blaga, M. C. Crasmareanu, The geometry of product conjugate connections , An.Univ. Ioan Cuza din Iasi, seria Matematica, tom LIX , Fasc. 1, (2013), 73–84. oliations induced by metallic structures [3] A. M. Blaga, A. 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