Almost contact metric and metallic Riemannian structures

. The metallic structure is a fascinating topic that continually gen- erates new ideas. In this work, new metallic manifolds are constructed starting from both almost contact metric manifolds and we obtain some important no- tions like the metallic deformation. We give a concrete example to con(cid:133)rm this construction. important role in establishing a relationship between mathematics and architecture.


Introduction
Manifolds equipped with certain di¤erential-geometric structures possess rich geometric structures and such manifolds and relations between them have been studied widely in di¤erential geometry. Indeed, almost complex manifolds, almost contact manifolds and almost product manifolds and relations between such manifolds have been studied extensively by many authors.
The di¤erential geometry of the Golden on Riemannian manifolds is a popular subject for mathematicians. In 2007, Hreţcanu [12] introduced the Golden structure on manifolds and in [14] the geometry of the golden structure on manifolds was studied. Now, Such manifolds have been studied by various authors (see [3,5,11,15,16]). Later, the author in [3] gave a set of techniques to construct many compatible well-known structures on a Riemannian manifold, starting from a Golden Riemannian manifold. And also he established in [4] an interesting class of almost Golden Riemannian manifolds such as the s-Golden manifolds.
As generalization of the Golden mean, the metallic means family appear in 1997 by Vera W. de Spinadel (see [10]) which contains the silver mean, the bronze mean, the copper mean and the nickel mean, etc. The metallic mean family plays an important role in establishing a relationship between mathematics and architecture.
For example, silver and golden mean can be seen in the sacred art of India, Egypt, China, Turkey and di¤erent ancient civilizations. Now, there are also several recent works in this direction [13,14,7] and others. Recently, a new type of structure on a di¤erentiable manifold is studied in [9] and the relation between metallic structure and almost quadratic '-structure is considered in [17].
Here we show that there exists a correspondence between the metallic Riemannian structures and the almost contact metric structures. This text is organized in the following way: Section 2 is devoted to the background of the structures which will be used in the sequel. In Section 3, starting from an almost contact metric structures we de…ne metallic Riemannian structures and we investigate conditions for those structures being integrable and parallel then we give an example to con…rm these latter properties. In Section 4, we give the notion of metallic transformation and we use it for some questions of the characterization of certain geometric structures. The Section 5 is devoted to give a generalization of the notion of metallic transformation which deduces the particular known cases. In the last Section, we give an open question where we propose the …rst step to study the reverse, i.e. the construction of an almost contact metric structure starting from a metallic Riemannian structure.

Review of needed notions
In this section, we give a brief information for metallic Riemannian manifolds and almost contact metric manifolds. We note that throughout this paper all manifolds and bundles, along with sections and connections, are assumed to be of class C 1 .
Let (M; g) be a Riemannian manifold. We present metallic Riemannian manifolds following [14]. A (p; q)-metallic structure on M is a polynomial structure of second degree given by a (1; 1)-tensor …eld which satis…es 2 = p + qI; where I is the identity transformation and p, q are …xed integers such that x 2 px q = 0 has a positive irrational root p;q . The number p;q is usually named a member of the metallic family. These numbers, denoted by: are also called (p; q)-metallic numbers. For example, we can talk about Golden structure if p = 1, q = 1 when the 1;1 is exactly the golden ratio = 1+ , or about the silver structure (p = 2, q = 1, 2;1 = 1 + p 2), the bronze structure (p = 3, q = 1, 3;1 = 3+ p 13 2 ) , the nickel structure (p = 1, q = 3, 1;3 = 1+ p 13 2 ) , the copper structure (p = 1, q = 2, 1;2 = 2). The above numbers are closely related with di¤erent mathematical domains as dynamical systems, quasicristales, theory of Cantorial fractal-like microspace-time.
For the Riemannian manifold (M; g) endowed with the (p; q)-metallic structure, we say that the metric g is -compatible and that M is a Riemannian metallic manifold [14], if g( X; Y ) = g(X; Y ); (3) for all X, Y vectors …els on M . If we substitute X into X in (3), equation (3) may also written as Here, we can show that such a metric always exists on a manifold with a metallic structure . Proof. Let h be any Riemannian metric on M and de…ne g by and check the details.
Note from Proposition 3:2 of [14] that every almost product structure J induces two metallic structures on M given as follows: is an almost product structure on M . Conversely, every metallic structure on M induces two almost product structures on M given as follows: For a metallic manifold (M; ; g) and the associated almost product J, it is easy to see that g(JX; Y ) = g(X; JY ); (6) for every tangent vector …elds X on M .
In order that the Golden structure is integrable, it is necessary and su¢ cient that it is possible to introduce a torsion-free a¢ ne connection r with respect to which the structure tensor is covariantly constant. Also, we know that the integrability of is equivalent to the vanishing of the Nijenhuis tensor N [14], The link between the Nihenjuis tensors and J is given by which show that the metallic structure is integrable if and ony if the associated almost product J is integrable. An odd-dimensional Riemannian manifold (M 2n+1 ; g) is said to be an almost contact metric manifold if there exist on M a (1; 1) tensor …eld ', a vector …eld (called the structure vector …eld) and a 1-form such that for any vectors …elds X,Y on M . In particular, in an almost contact metric manifold we also have ' = 0 and ' = 0: Such a manifold is said to be a contact metric manifold if where (X; On the other hand, the almost contact metric structure of M is said to be normal if for any X and Y vectors …elds on M , where ['; '] denotes the Nijenhuis torsion of ', given by An almost contact metric structure ('; ; ; g) on M is said to be: where d denotes the exterior derivative.

Induced Metallic structures by almost contact structures
In this section, starting from an almost contact metric structure we de…ne a metallic Riemannian structure and we investigate conditions for those structures being integrable and parallel.
Theorem 4. Every almost contact metric structure ('; ; ; g) on a (2n + 1)dimensional Riemannian manifold (M; g) induces only two metallic structures on (M; g), given as follows: where is the unique eigenvector of 1 and 2 associated with p;q = p p;q and p;q respectively.
Proof. We try to write the metallic structure i with i 2 f1; 2g de…ned on a (2n+1)dimensional Riemannian manifold (M; g), using almost contact metric structure ('; ; ; g), in the form = a i I + b i , where a i and b i are non-zero constant.
and using formula (1) with 1 = p;q and 2 = p;q , we obtain the formulas (16). Moreover, we have for every i 2 f1; 2g and for every tangent vectors …elds X and Y on M .
On the other hand, suppose that there exist another metallic structure on M induces by the almost contact metric structure ('; ; ; g) denoted by and admits as the unique eigenvector associated with p;q ( resp. p;q ) then, we have 2 = p + qI; = p;q (resp: = p;q ): First, note that for all i 2 f1; 2g we have i = i ; and using (1) and (17)  Proof. Since the proof of the following proposition is obvious, we don't give the proof of it.
Using formula (4), we get the following: Proposition 7. Every almost contact metric manifold (M 2n+1 ; '; ; ; g) induces four almost product structures on (M; g), given as follows: We note that through out this paper, we shall be setting = p;q I + (p 2 p;q ) : Observe that, 2 p;q = p p;q + q; p;q + p;q = p and p;q : p;q = q: We know that the metallic structure is integrable ( i.e. N = 0 ) if and only if the almost product J is integrable ( i.e. N J = 0) with So, for all X; Y vectors …elds on M and using (18), we get witch give the following theorem: Applying (22) in (21) we get d (X; Y ) = 0; for all X and Y vectors …elds tangent to M .
Proposition 11. Let (M 2n+1 ; ; g) be a metallic Riemannian manifold induced by the almost contact metric manifold (M 2n+1 ; '; ; ; g). If r is the Levi-Cevita connection then for all X and Y vectors …elds tangent to M we have (r X )Y = (p 2 p;q ) g(r X ; Y ) + (Y )r X : and using formula (19), the proof is direct.
On the other hand, we know that the integrability of is equivalent to the existence of a torsion-free a¢ ne connection with respect to which the equation r = 0 holds. Now we shall introduce another possible su¢ cient condition of the integrability of metallic structures on Riemannian manifolds. Proof. The necessity was observed above (see (24)). For the su¢ ciency, it su¢ ces to replace Y by in (24). Remark 13. If ('; ; ; g) is a cosymplectic structure then ( ; g) is a parallel metallic Riemannian structure.
We is also an almost contact metric structure.
We refer to this construction as metallic deformation.
Remark 17. Note that the metallic transformation preserve the structure cosymplectic for all two positive integers p and q and the Kenmotsu structure only for q = p + 1 but the Sasakian structure is never preserved.
A straightforward computation yields the following proposition: 1022 G HERICI BELDJILALI Proposition 18. If ( ; g) be a metallic Riemannian structure induced by the almost contact metric structure ('; ; ; g), then the structure g(X; Y ) = g( n X; n Y ) = 2n p;q g + 2n p;q (p p;q ) 2n : for any integer number n, is also an almost contact metric structure.

Generalized D-homothetic transformation
Let (M; '; ; ; g) be an almost contact metric manifold with dimM = 2n + 1. The equation = 0 de…nes a 2n-dimensional distribution D on M . By an 2nhomothetic deformation or D-homothetic deformation [22] we mean a change of structure tensors of the form where a is a positive constant. If (M; '; ; ; g) is a contact metric structure with contact form , then (M ; '; ; ; g) is also a contact metric structure [22].
This idea works equally well for almost contact metric structures. The deformation' is again an almost contact metric structure where f and h are two non-zero functions on M . From the theorem (16), we can deduce the following proposition:

Special cases:
For h = f , we get the conformal transformation [21].
For h = f 2 and f = constant, we get the deformation of Tanno [22]. For h = 1, we get the deformation of Marrero [19]. For f = 1 , we get the D-isometric [2].

Open problem
Finally, we propose the …rst steps to construct an almost contact metric structure from a metallic Riemannian structure. Let (M 2n+1 ; ; g) be a metallic Riemannian manifold and be the unique eigenvector of associated with p;q = p p;q ( resp. p;q ) which give = p;q ( resp. = p;q ) and let be the g-dual of i.e. (X) = g(X; ) for all vector …eld X on M such that ( ) = 1.
Proposition 20. The metallic structure admits the following expression: = p;q I + (p 2 p;q ) ; resp: = p;q I + (p 2 p;q ) ; (27) Proof. We try to write the metallic structure in the form = aI + b , where a; b 2 R . Thus 2 = a 2 I + b(a + p;q ) ; on the other hand, we have p + qI = (ap + q)I + pb ; using formulas (1) we obtain the formulas (27).
One can construct on M 2n+1 an almost contact metric structure ('; ; ; g) starting from a metallic Riemannian structure and study its nature taking into account the two parameters p and q.