Generalized metallic structures

We study the properties of a generalized metallic, a generalized product and a generalized complex structure induced on the generalized tangent bundle of $M$ by a metallic Riemannian structure $(J,g)$ on $M$, providing conditions for their integrability with respect to a suitable connection. Moreover, by using methods of generalized geometry, we lift $(J,g)$ to metallic Riemannian structures on the tangent and cotangent bundles of $M$, underlying the relations between them.


Preliminaries
On a smooth manifold M , besides the almost complex, almost tangent, almost product structures, etc., some other polynomial structures can be considered as C ∞ -tensor fields J of (1, 1)-type which are roots of the algebraic equation where I is the identity operator on the Lie algebra of vector fields on M .In particular, if Q(J) := J 2 −pJ −qI, with p and q positive integers, its solution J will be called metallic structure [2].The name is motivated by the fact that the (p, q)metallic number introduced by Vera W. de Spinadel [8] is precisely the positive root of the quadratic equation x 2 − px − q = 0, namely σ p,q := p + p 2 + 4q 2 .For example: if p = q = 1 we get the golden number σ = 1 + √ 5 2 ; if p = 2 and q = 1 we get the silver number σ 2,1 = 1 + √ 2; if p = 3 and q = 1 we get the bronze number σ 3,1 = 3 + √ 13 2 ; if p = 1 and q = 2 we get the copper number σ 1,2 = 2; if p = 1 and q = 3 we get the nickel number , and so on.
We shall briefly recall the basic notions of metallic (Riemannian) geometry.

Definition 1.1 ([3]
).A metallic structure J on M is an endomorphism J : T M → T M satisfying J 2 = pJ + qI, (1.1) for some p, q ∈ N * .The pair (M, J) is called a metallic manifold.Moreover, if a Riemannian metric g on M is compatible with J, that is g(JX, Y ) = g(X, JY ), for any X, Y ∈ C ∞ (T M ), we call the pair (J, g) a metallic Riemannian structure and (M, J, g) a metallic Riemannian manifold.
The concept of integrability for a metallic structure is defined in the classical manner.
Definition 1.2.A metallic structure J is called integrable if its Nijenhuis tensor field It is known [3] that an almost product structure F on M induces two metallic structures: and, conversely, every metallic structure J on M induces two almost product structures: where σ p,q = p + p 2 + 4q 2 is the metallic number, for p, q ∈ N * .In particular, if the almost product structure F is compatible with a Riemannian metric g, then (J + , g) and (J − , g) are metallic Riemannian structures.
The analogue concept of locally product manifold is considered in the context of metallic geometry.

Definition 1.3 ([1]
).A metallic Riemannian manifold (M, J, g) is called locally metallic if J is parallel with respect to the Levi-Civita connection ∇ of g, that is ∇J = 0.
In the following, we shall extend the definition of a metallic structure for any real numbers p and q.In this way, we also include some other well-known structures; for instance, if (p, q) ∈ {(0, −1), (0, 0), (0, 1), (1, 0)}, the solution of (1.1) would yield an almost complex, an almost tangent, an almost product and a J(2, 1)-structure, respectively.

Generalized structures induced by metallic structures
Let T M ⊕ T * M be the generalized tangent bundle of a smooth manifold M .
for some real numbers p and q.
For a linear connection ∇ on M , we consider the bracket [ 2.1.Generalized metallic structure induced by (J, g).Let (J, g) be a metallic Riemannian structure on M such that J 2 = pJ + qI, p, q ∈ R. If we denote by g : T * M → T M the inverse of the isomorphism g : T M → T * M , g (X) := i X g, from the g-symmetry of J we have g • J * = J • g and g • J = J * • g , where (J * α)(X) := α(JX).Also notice that J * is a metallic structure too, namely, (J * ) 2 = pJ * +qI, and we easily get that g On T M ⊕ T * M we consider the Riemannian metric Definition 2.3.A pair ( Ĵ, ĝ) of a generalized metallic structure Ĵ and a Riemannian metric ĝ such that Ĵ is ĝ-symmetric is called generalized metallic Riemannian structure.
Proposition 2.4.The generalized metallic structure Ĵm induced by the metallic Riemannian structure (J, g) on M is ∇-integrable if and only if J is integrable and Proof.We have: then a direct computation gives: In particular, if ∇ is a torsion free J-connection, then Ĵm is ∇-integrable.
Let ∇ g be the Levi-Civita connection of g and define a linear connection D on M by D := ∇ g + F , where F is a (1, 2)-type tensor field such that Define the connection D on T M ⊕ T * M by [7]: Let n be the dimension of M and assume that q = 0. Denote by {x 1 , . . ., x n } the local coordinates on M and let {X 1 , . . ., X n } be the corresponding local frame for T M .Following [4] we define: where ω is a 1-form on M and we use Einstein's convention of summation.We immediately have that g(F (X i , X j ), X r )+g(X j , F (X i , X r )) = 0, for all i, j, r; therefore, Dg = 0, for any 1-form ω.Moreover, the torsion of D is given by Lemma 2.5.T D satisfies the following properties: Proof.From a direct computation we get Recently, C. Karaman [4] constructed metallic semi-symmetric metric J-connections D on locally decomposable metallic Riemannian manifolds (M, J, g).These connections satisfy: In particular, we can state the following: Proposition 2.6.Let (M, J, g) be a locally decomposable metallic Riemannian manifold and let D be a metallic semi-symmetric metric J-connection.Then Ĵm is D-integrable.
Remark 2.9.A metallic diffeomorphism f between two metallic manifolds (M 1 , J 1 ) and (M 2 , J 2 ) naturally induces an isomorphism f between their generalized tangent bundles defined by , which preserves the generalized metallic structures Ĵi,m := In particular, if f : M → M is a diffeomorphism which preserves the metallic structure J, then f can be defined by which coincides with the generalized metallic structure Ĵm when J = f * .In this case, J is invertible and J −1 = 1 q J − p q I, for q = 0.
2.2.Generalized product structure induced by (J, g).Let (J, g) be a metallic Riemannian structure on M such that J 2 = pJ + qI, p, q ∈ R. Then Ĵp := A direct computation gives the following.
Proposition 2.10.The generalized product structure Ĵp induced by the metallic Riemannian structure (J, g) on M is ∇-integrable if and only if the following conditions are satisfied: , where we denoted g by g and the exterior differential associated to ∇ acting on g by Proposition 2.11.Let (M, J, g) be a locally metallic Riemannian manifold.Then Ĵp is ∇-integrable, for ∇ the Levi-Civita connection of g.

Proof.
From the previous proposition, we have that the generalized product structure Ĵp is ∇-integrable if and only if the following conditions are satisfied: In particular, if ∇J = 0, then Ĵp is ∇-integrable.
Rev. Un.Mat.Argentina, Vol.61, No. 1 (2020) Definition 2.12.A generalized product structure Ĵ on M is called anti-pseudocalibrated if it is (•, •)-anti-invariant and the bilinear symmetric form defined by (•, Ĵ•) on T M is non-degenerate, where is the natural symplectic structure on T M ⊕ T * M .
Proposition 2.14.Let Ĵp be the generalized product structure defined by the metallic Riemannian structure (J, g) on M .Then Proof.Locally we can write 2G in block matrix form as: As J is g-symmetric, pointwise, we can take g = I and J = Λ the diagonal matrix with eigenvalues λ 1 , . . ., λ n which are solutions of the metallic equation λ 2 − pλ − q = 0. Then we get: In order to compute the indices of 2G, we can use the Gauss-Lagrange algorithm and by elementary operations on rows and columns of the matrix we get the form hence 2G has n positive and n negative eigenvalues and the proof is complete.Remark 2.16.Starting with a metallic structure on a manifold, with minimal restrictions on p and q, some other generalized metallic structures on its generalized tangent bundle can be constructed as follows.
The metallic structure J on M induces two almost product structures on M : the almost product structures F ± induce two generalized product structures on T M ⊕ T * M : and the generalized product structures F ± induce two generalized metallic structures on T M ⊕ T * M : where The metallic structure J on M induces a generalized product structure on T M ⊕ T * M : and the generalized product structure Ĵp induces two generalized metallic structures on T M ⊕ T * M : 2.3.Generalized complex structure induced by (J, g).Let (J, g) be a metallic Riemannian structure on M such that J 2 = pJ + qI, p, q ∈ R. Then Ĵc := J −(I + J 2 ) g g −J * is a generalized complex structure on M , that is, Ĵ2 c = −I [6].A direct computation gives the following.
Proposition 2.17.The generalized complex structure Ĵc induced by the metallic Riemannian structure (J, g) on M is ∇-integrable if and only if the following conditions are satisfied: , where we denoted g by g and the exterior differential associated to ∇ acting on g by Proposition 2.18.Let (M, J, g) be a locally metallic Riemannian manifold.Then Ĵc is ∇-integrable, for ∇ the Levi-Civita connection of g.

Proof.
From the previous proposition, we have that the generalized complex structure Ĵc is ∇-integrable if and only if the following conditions are satisfied: and the bilinear symmetric form defined by (•, Ĵ•) on T M is non-degenerate and positive definite, where is the natural symplectic structure on T M ⊕ T * M .
Definition 2.22.A pair ( Ĵc , Ĵp ) of a generalized complex structure and a generalized product structure is called generalized complex product structure if Ĵc Ĵp = − Ĵp Ĵc .

Metallic structures on tangent and cotangent bundles
3.1.Metallic structure on the tangent bundle.Let (M, J, g) be a metallic Riemannian manifold and let ∇ be a linear connection on M .∇ defines the decomposition into the horizontal and vertical subbundles of T (T M ): Let π : T M → M be the canonical projection and π * : T (T M ) → T M be the tangent map of π.If a ∈ T M and A ∈ T a (T M ), then π * (A) ∈ T π(a) M and we denote by χ a the standard identification between T π(a) M and its tangent space T a (T π(a) M ).
Let Ψ ∇ : T M ⊕ T * M → T (T M ) be the bundle morphism defined by Ψ ∇ (X + α) := X H a + χ a ( g α), where a ∈ T M and X H a is the horizontal lifting of X ∈ T π(a) M .Let x 1 , . . ., x n be local coordinates on M , let x1 , . . ., xn , y 1 , . . ., y n be respectively the corresponding local coordinates on T M , and let X 1 , . . ., X n , ∂ ∂y 1 , . . ., ∂ ∂y n be a local frame on T (T M ), where X i = ∂ ∂ xi .We have: Rev. Un.Mat.Argentina, Vol.61, No. 1 (2020) where i, k, l run from 1 to n and Γ k il are the Christoffel symbols of ∇.Let Ψ ∇ : T M ⊕ T * M → T (T M ) be the bundle morphism defined before (which is an isomorphism on the fibres).In local coordinates, we have the following expressions: Let ( Ĵm , ĝ) be the generalized metallic structure defined in the previous section.The isomorphism Ψ ∇ allows us to construct a natural metallic structure Jm and a natural Riemannian metric ḡ on T M in the following way.
We define Jm : and the Riemannian metric ḡ on T M by ḡ := ((Ψ ∇ ) −1 ) * (ĝ).In local coordinates, we have the following expressions for Jm and ḡ: Computing the Nijenhuis tensor of Jm , we get the following: Therefore we can state the following.Proposition 3.2.Let (M, J, g) be a flat locally metallic Riemannian manifold.If ∇ is the Levi-Civita connection of g, then ( Jm , ḡ) is an integrable metallic Riemannian structure on T M .

3.2.
Metallic structure on the cotangent bundle.Let (M, J, g) be a metallic Riemannian manifold and let ∇ be a linear connection on M .∇ defines the decomposition into the horizontal and vertical subbundles of T (T * M ): Let π : T * M → M be the canonical projection and π * : T (T * M ) → T M be the tangent map of π.If a ∈ T * M and A ∈ T a (T * M ), then π * (A) ∈ T π(a) M and we denote by χ a the standard identification between T * π(a) M and its tangent space T a (T * π(a) M ).Let Φ ∇ : T M ⊕ T * M → T (T * M ) be the bundle morphism defined by [5]: where i, k, l run from 1 to n and Γ k il are the Christoffel symbols of ∇.
Let Φ ∇ : T M ⊕ T * M → T (T * M ) be the bundle morphism defined before (which is an isomorphism on the fibres).In local coordinates, we have the following expressions: Let ( Ĵm , ĝ) be the generalized metallic structure defined in the previous section.The isomorphism Φ ∇ allows us to construct a natural metallic structure Jm and a natural Riemannian metric g on T * M in the following way.
We Proof.From the definition it follows that J2 m = p Jm + qI and g( Jm X, Y ) = g(X, Jm Y ), for any X, Y ∈ C ∞ (T (T * M )).
In local coordinates, we have the following expressions for Jm and g: Computing the Nijenhuis tensor of Jm , we get the following: Therefore we can state the following.
), where a ∈ T * M and X H a is the horizontal lifting of X ∈ T π(a) M .Let x 1 , . . ., x n be local coordinates on M , let x1 , . . ., xn , y 1 , . . ., y n be respectively the corresponding local coordinates on T * M and let {X 1 , . . ., X n , * M ), where X i = ∂ ∂ xi .We have: