METALLIC CONJUGATE CONNECTIONS

Properties of metallic conjugate connections are stated by pointing out their relation to product conjugate connections. We define the analogous in metallic geometry of the structural and the virtual tensors from the almost product geometry and express the metallic conjugate connections in terms of these tensors. From an applied point of view we consider invariant distributions with respect to the metallic structure and for a natural pair of complementary distributions, the above structural and virtual tensors are expressed in terms of O’Neill–Gray tensor fields.


Introduction
Besides the very well known almost complex, almost tangent, and almost product structures on a differentiable manifold M , some other polynomial structures naturally arise as C ∞ -tensor fields J of type (1,1) which are roots of the algebraic equation Q(J) := J n + a n J n−1 + · · · + a 2 J + a 1 I X(M ) = 0, where I X(M ) is the identity map on the Lie algebra of vector fields on M . In particular, if the structure polynomial is Q(J) := J 2 − pJ − qI X(M ) , with p and q positive integers, its solution J will be called metallic structure [11]. The name is motivated by the fact that, for different values of p and q, the (p, q)-metallic number introduced by Vera W. de Spinadel [14] 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: • for p = q = 1 we get the golden number σ = 1 + √ 5 2 which is the limit of the ratio of two consecutive Fibonacci numbers [13]; • for p = 2 and q = 1 we get the silver number σ 2,1 = 1 + √ 2 which is the limit of the ratio of two consecutive Pell numbers [9]; • for p = 3 and q = 1 we get the bronze number σ 3,1 = 3 + √ 13 2 which plays an important role in the study of dynamical systems and quasicrystals; • for p = 1 and q = 2 we get the copper number σ 1,2 = 2; • for p = 1 and q = 3 we get the nickel number σ 1,3 = 1 + √ 13 2 , and so on.
Notice that if J is a metallic structure on M with J 2 = pJ + qI X(M ) , thenJ := aJ + bI X(M ) is also a metallic structure withJ 2 =pJ +qI X(M ) , for a = 2σp ,q −p 2σ p,q − p In this setting, we shall study the properties of the conjugate connections (by a metallic structure), express their virtual and structural tensor fields and study their behavior on invariant distributions. Finally, taking into account the connection between metallic and almost product structures stated in [12], we shall analyze the impact of the duality between the metallic and almost product structures on metallic and product conjugate connections.
Let M be a smooth, n-dimensional manifold and denote by C ∞ (M ) the algebra of smooth real functions on M , by X (M ) the Lie algebra of vector fields on M , and by T r s (M ) the C ∞ (M )-module of tensor fields of (r, s)-type on M . Usually X, Y , Z, . . . will be vector fields on M and if T → M is a vector bundle over M , then Γ(T ) denotes the C ∞ -module of sections of T , e.g. Γ(T M ) = X(M ).
Consider C(M ) the set of all linear connections on M . Since the difference of two linear connections is a tensor field of (1, 2)-type, it results that C(M ) is a C ∞ (M )-affine module associated to the C ∞ (M )-linear module T 1 2 (M ). Recall the concept of metallic (Riemannian) geometry: for p, q ∈ N * , where I X(M ) is the identity operator on X(M ). The pair (M, J) is 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 ∈ X(M ), we call the pair (g, J) metallic Riemannian structure and (M, g, J) metallic Riemannian manifold.
It was shown in [12] that the powers of J satisfy where {g n } n∈N * is the generalized secondary Fibonacci sequence defined by g n+1 = pg n + qg n−1 , n ≥ 1, with g 0 = 0, g 1 = 1 and p, q ∈ N * . Remark 1.1. i) The idea of introducing a metallic structure [12] as a solution of the equation J 2 − pJ − qI X(M ) = 0 for p, q ∈ N * was based on the fact that the positive root of the corresponding equation x 2 −px−q = 0 is precisely a metallic number for p, q ∈ N * . By letting p and q be any natural number, we also include some other well-known structures; for instance, if (p, q) ∈ {(0, 0), (0, 1), (1, 0)}, the solution of (1.1) would yield an almost tangent, an almost product, and a J(2, 1)-structure, respectively. ii) Concerning the inheritance of the metallic structure on submanifolds, Hreţcanu and Crasmareanu proved in [12] that a metallic structure on a metallic Riemannian manifold M induces a metallic structure on every invariant submanifold of M (i.e., on those submanifolds N of M for which J(T x N ) ⊂ T x N , for any x ∈ N ) and illustrate this on a product of spheres in an Euclidean space.
Fix now J a metallic structure on M and define the associated linear connections as follows. The concept of integrability is defined in the classical manner: Necessary and sufficient conditions for the integrability of a polynomial structure J whose characteristic polynomial has only simple roots were given by Vanzura in [15], who proved that if there exists a symmetric linear J-connection ∇, then the structure J is integrable.
In order to find a measure of how far away a linear connection is from being in C J (M ) we introduce in the next section the notion of metallic conjugate connection. The present paper deals with the study of these connections and then it is the fifth in a series containing [4], [5], [3] and [6]. We remark that Bejan and Crasmareanu [1] studied the conjugate connections with respect to a quadratic endomorphism as a generalization of almost complex and almost product cases. An important tool in our work is provided by the pair (structural tensor, virtual tensor) defined for the almost product geometry in [5] and considered here in the last part of Section 1. After treating in Section 2 the duality between the metallic and the product conjugate connections, from an applied point of view we consider in Section 3 the J-invariant distributions.

Metallic conjugate connection
Let ∇ be a linear connection on the metallic manifold (M, J) and define the metallic conjugate connection of ∇ by: Recall that given a linear connection ∇, one defines its torsion as the (1, 2)-tensor field for any X, Y , Z ∈ X(M ). We call ∇ a symmetric (respectively, flat) connection if T ∇ = 0 (respectively, R ∇ = 0).
is a metallic Riemannian manifold, then: is also a g-metric connection (belonging to C J (M )).
which yields the claimed formula. (2) and the expression of curvature becomes (4) Using that g(JX, Y ) = g(X, JY ) and g(JX, JY ) = pg(X, JY ) + qg(X, Y ) we obtain: Recall that a linear connection ∇ is called quarter-symmetric connection [10] if its torsion is of the form where ω is a 1-form and T is a (1, 1)-tensor field.
A natural generalization of the case ∇ ∈ C J (M ) is given by: Proposition 2.2. Let ∇ be a symmetric linear connection. Assume that ∇J = η ⊗ J n , for n ∈ N, where η is a 1-form. Then ∇ (J) = ∇ + η ⊗ J n+1 and it is a quarter-symmetric connection.
and the result follows from d ∇ J = 0.
Definition 2.5. We say that (g, J, ∇) is a special metallic structure if (g, J) is a metallic Riemannian structure and ∇ is a special metallic connection.
We remark that for a given pair (∇, J), because of the duality between the metallic and the almost product structures (see Section 2), there always exists a Riemannian metric g such that (g, J, ∇) is a special metallic structure. Let R 5 = R 2 ×R 3 be the Euclidean space with the metric given by the Euclidean scalar product g = ·, · and denote by (x 1 , x 2 , y 1 , y 2 , y 3 ) the local coordinates in R 5 .
Example 2.2. In a similar way, one can construct special metallic structures on the Euclidean spaces of even dimension.
Let R 4 = R 2 ×R 2 be the Euclidean space with the metric given by the Euclidean scalar product g = ·, · and denote by (x 1 , x 2 , y 1 , y 2 ) the local coordinates in R 4 . In a similar manner as in Example 2.1, we can check that for given p, q ∈ N * , the (1, 1)-tensor fields J 0 : X(R 4 ) → X(R 4 ) defined by The last subject of this section treats two tensor fields associated to a metallic structure. The paper [5] introduces the structural and virtual tensor fields of an almost product structure. Turning into our framework, let us consider for a pair (∇, J) the following tensor fields of (1, 2)-type: 1) the structural tensor field

2) the virtual tensor field
It results that Y )) and . The importance of these tensor fields for our study is underlined by the following relation: 3. If the linear connection ∇ satisfies ∇J = η ⊗ J n , for n ∈ N, where η is a 1-form, then the structural and the virtual tensor fields have the expressions Example 2.4. Concerning the behavior of ∇ (.) for families of metallic structures, remark that if J 1 and J 2 are two metallic structures with J 2 1 = pJ 1 + q 1 I X(M ) and J 2 2 = pJ 2 +q 2 I X(M ) , then J := J 1 +J 2 is a metallic structure with J 2 = pJ +qI X(M ) if and only if J 1 J 2 + J 2 J 1 = (q − q 1 − q 2 )I X(M ) ; in this case:

The duality of metallic and product conjugate connections
In [12] Hreţcanu and Crasmareanu proved that any metallic structure J induces two almost product structures: and for any given pair (p, q) ∈ N * × N * , any almost product structure E determines two metallic structures: Then ∇E ± = ± 2 2σ p,q − p ∇J and ∇J ± = ± 2σ p,q − p 2 ∇E respectively. Hence, ∇ is a J-connection if and only if ∇ is an E-connection.
We are interested in finding the relation between the conjugate connections associated to them.
We showed in [5] that for an almost product structure E, the structural and virtual tensor fields satisfy If E is an almost product structure on M and J ± are the metallic structures given by (3.2), then Then JY ∈ Γ(D l ) and for any X ∈ Γ(T M ), we have A more general notion is given by geodesic invariance [7]. The distribution D is ∇-geodesically invariant if for every geodesic γ : [a, b] → M of ∇ withγ(a) ∈ D γ(a) it follows thatγ(t) ∈ D γ(t) , for any t ∈ [a, b]. A necessary and sufficient condition for a distribution D to be ∇-geodesically invariant is given in [7]: for any X and Y ∈ Γ(D), the symmetric product X : Y := ∇ X Y + ∇ Y X to belong to Γ(D) or equivalently, for any X ∈ Γ(D) to have ∇ X X ∈ Γ(D).
The following result is a direct consequence of definitions: We remark that E := l − m is an almost product structure and both D l and D m are E-invariant. The product conjugate connection of ∇ is With X → mX and Y → mY in the first relation above it follows that l[mX, mY ] = 0 and the change X → lX and Y → mY in the second relation yields m[lX, lY ] = 0.
We have: 1) ∇ restricts to D l means m(∇ X lY ) = 0 and l(∇ X lY ) = ∇ X lY , 2) ∇ restricts to D m means l(∇ X mY ) = 0 and m(∇ X mY ) = ∇ X mY . A straightforward computation gives that ∇ (E) from (4.1) restricts to D l and D m . Moreover, if ∇ restricts to both D l and D m , then and so ∇ ∈ C E (M ). Let us remark that the above connection (4.2) is exactly the Schouten connection of the pair (l, m) [8]: ∇ X Y = l(∇ X lY ) + m(∇ X mY ).
We shall express the structural and the virtual tensor fields of the almost product structure E in terms of the projectors l, m as follows: