Inspired by the metallic means family, Hretcanu and Cras- mareanu [14] constructed a new structure on a Riemannian manifold and named it a metallic structure

https://doi.org/10.5831/HMJ.2020.42.1.63

OPERATORS ON METALLIC RIEMANNIAN STRUCTURES

Has¸im C¸ ayir

Abstract. In this paper firstly, Some properties were given about metallic Riemannian structure on (1, 1)−tensor bundle. Secondly, the Tachibana and Vishnevskii operators were applied to vertical and horizontal lifts with respect to the metallic Riemannian struc- ture on (1, 1)−tensor bundle, respectively.

1. Introduction

Let M be n−dimensional manifold. The metallic means family or metallic proportions, was introduced by de Spinadel in [8, 9, 10, 11].

For two positive integers p and q, the positive solition of the equation x²− px − q = 0 is named members of the metallic means family. All the members of metallic means family are positive quadratic irrational num- bers σ_p,q = ^p+

√

p²+4q

2 . These numbers σ_p,qare also called (p, q)−metallic numbers. Inspired by the metallic means family, Hretcanu and Cras- mareanu [14] constructed a new structure on a Riemannian manifold and named it a metallic structure. Indeed, a metallic structure is a poly- nomial with the structural polynomial Q(J ) = J²− pJ − qI. Polynomial structures on a manifold were defined in [13]. A polynomial structure F of degree d on a connected manifold M means that a (1, 1)−tensor field F satisfies the following algebraic polynomial equation:

Q(F ) = F^d+ a₁F^d−1+ ... + a_d−1F + a_dI = 0,

where a₁, a₂, ..., a_d are real numbers and I is the identity tensor of type (1, 1).

Received April 2, 2019. Revised May 18, 2019. Accepted May 20, 2019.

2010 Mathematics Subject Classification. 15A72, 53A45, 47B47, 53C15.

Key words and phrases. Metallic Riemannian structure, (1, 1)−tensor bundle, Tachibana operators, Vishnevskii operators, Horizontal lift, Vertical lift.

Given a Riemannian manifold (M, g) endowed with the metallic struc- ture J , then the triple (M, J, g) is named a metallic Riemannian manifold if

(1) g(J X, Y ) = g(X, J Y )

or equivalently

g(J X, J Y ) = g(J²X, Y ) = g((pJ + qI)X, Y ) = pg(J X, Y ) + qg(X, Y ) for all vector fields X and Y on M [14]. The Riemannian metric (1) is referred to as J −compatible or pure metric [16, 24, 25]. In addition, the Riemannian manifolds and the tangent bundles studyed a lot of authors [1, 2, 3, 4, 15, 16, 17, 18, 19, 20, 23] too.

Theorem 1.1. [12] Let M be a metallic Riemannian manifold equipped with a metallic structure J and a Riemannian metric g. Then:

a): J is integrable if φ_Jg = 0,

b): the condition φJg = 0 is equivalent to ∇J = 0, where ∇ is the Levi-Civita connection of g.

Proposition 1.2. [12, 14] If J is a metallic structure on M , then F±= ±( 2

2σp,q− pJ − p 2σp,q− pI)

are two almost product structures on M. Conversely, every almost prod- uct structure F on M induces two metallic structures on M , given as follows:

J± = p

2I ± (2σ_p,q− p 2 )F.

Let M be n−dimensional Riemannian manifold with a Riemannian metric g and denote by π : T₁¹(M ) −→ M its (1, 1)−tensor bundle with fibers the (1, 1)−tensor spaces to M . Then T₁¹(M ) is an n + n²−dimensional smooth manifold and some local charts induced nat- urally from local charts on M may be used. Namely, a system of local coordinates (U ; x^j) in M induces on T₁¹(M ) a system of local coordinates (π⁻¹(U ); x^j, x^¯j = tⁱ_j) j = 1, ..., n, ¯j = n + 1, ..., n + n², J = 1, ..., n + n², where (tⁱ_j) are the Cartesian coordinates in each (1, 1)−tensor space T_{1(P )}¹ M at P ∈ M with respect to the natural base.

Let X = X^{i ∂}_∂xi and A = Aⁱ_j∂x^∂i ⊗ dx^j be the local expressions in U of a vector field X and a (1, 1)−tensor field A on M , respectively. Then

the vertical lift ^VA of A and the horizontal lift^HX of X are given, with respect to the induced coordinates, by [12]

A^V =

 A^{j V} A^{¯j V}



=

 0 Aⁱ_j



and

X^H =

 X^{j H} X^{¯j H}



=

 X^j

X^s(Γ^m_sjtⁱ_m− Γⁱ_smt^m_j )



where Γ^h_ij are the coefficients of the Levi-Civita connection ∇ of g.

The Sasaki type metric^Sg on T₁¹(M ) is defined by the following three equations:

Sg(A^V, B^V) = g(A, B),

Sg(A^V, Y^H) = 0,

Sg(X^H, Y^H) = g(X, Y ),

for any vector fields X and Y and (1, 1)−tensor fields A,B on M , where g(A, B) = g_itg^jlAⁱ_jB^t_l [12, 22].

The bracket operation of vertical and horizontal vector fields is given by the formulas

[^VA,^V B] = 0 , (2)

[^HX,^V A] = ^V(∇_XA),

[^HX,^HY ] = ^H[X, Y ] + (˜γ − γ)R(X, Y ),

where R denotes the curvature tensor field of the connection ∇, and

˜

γ − γ : ϕ → =¹₀(T₁¹(M )) is the operator defined by (˜γ − γ)ϕ =

 0

tⁱ_mϕ^m_j − t^m_j ϕⁱ_m



for any ϕ ∈ =¹₁(M ) [22].

The horizontal lifts, orthogonal to the fibers of T₁¹(M ). Let now E be a nowhere zero vector field on M . For any vector field X and covector field ˜E = g ◦ E on M , we define the vertical lift (X ⊗ ˜E)^V of X with respect to E. The map X → (X ⊗ ˜E)^V is a monomorphism. Hence, an n−dimensional C^∞ vertical distribution V^E is defined on T₁¹(M ).

Let V^⊥ be the distribution on T₁¹(M ), which is orthogonal to H and V^E. Then H, V^E and V^⊥ are mutually orthogonal distributions with

respect to the Sasaki type metric ^Sg. We define a (1, 1)−tensor field ˜J on T₁¹(M ) by [12]

(3)







J X˜ ^H = ^p₂X^H + (^2σp,q₂^−p)(X ⊗ ˜E)^V, J (X ⊗ ˜˜ E)^V = ^p₂(X ⊗ ˜E)^V + (^2σp,q₂^−p)X^H, J (A˜ ^V) = σ_p,q A^V,

for any vector field X and (1, 1)−tensor fields A on M , where ˜E = g ◦ E is a covector field on M . The restrictions of ˜J to H + V^E and V^⊥ are endomorphisms, and hence ˜J is a (1, 1)−tensor field on T₁¹(M ). It is easily see that ˜J²− p ˜J − qI = 0, i.e. ˜J is a metallic structure on T₁¹(M ).

We point out here and once that all geometric objects considered in this paper are supposed to be of class C^∞

2. Main Results

2.1. The Tachibana operators applied to vertical and hori- zontal lifts with respect to metallic Riemannian struc- ture.

Definition 2.1. Let ϕ ∈ =¹₁(M ), and =(M ) = P∞

r,s=0=^r_s(M ) be a tensor algebra over R. A map φ_ϕ |

r+si0:

∗

=(M ) → =(M ) is called a Tachibana operator or φϕ operator on M if

a) φϕ is linear with respect to constant coefficient, b) φϕ:

∗

=(M ) → =^r_s+1(M ) for all r and s,

c) φϕ(K⊗ L) = (φ^C _ϕK) ⊗ L + K ⊗ φϕL for all K, L ∈

∗

=(M ),

d) φ_ϕXY = −(L_Yϕ)X for all X, Y ∈ =¹₀(M ) where L_Y is the Lie derivation with respect to Y,

e)

(φϕXη)Y = (d(ıYη))(ϕX) − (d(ıY(ηoϕ)))X + η((LYϕ)X) (4)

= φX(ı_Yη) − X(ı_ϕYη) + η((L_Yϕ)X)

for all η ∈ =⁰₁(M ) and X, Y ∈ =¹₀(M ), where ıYη = η(Y ) = η⊗Y,^C

∗

=^r_s(M ) the module of all pure tensor fields of type (r, s) on M according to the affinor field ϕ [5, 6, 7, 17](see [21] for applied to pure tensor field).

Theorem 2.2. For LX the operator Lie derivation with respect to X, ˜J is a metallic structure on T₁¹(M ) defined by (3), φ_J˜the Tachibana

operator on M , we get the following formulas

i) φJ X˜ ^HY^H = (−p

2 + σp,q)(˜γ − γ)R(Y, X)

−(2σp,q− p

2 )((∇_YX) ⊗ ˜E +X ⊗ (g ◦ (∇_YE)) +(L_YX) ⊗ ˜E)^V ii) φJ X˜ ^H(Y ⊗ ˜E)^V = (p

2− σ_p,q)(Y ⊗ (g ◦ (∇_XE)))^V

−(2σ_p,q− p

2 )(∇_XY )^H iii) φJ (X⊗ ˜˜ E)^V(Y ⊗ ˜E)^V = (2σ_p,q− p

2 )((∇XY ) ⊗ ˜E +Y ⊗ (g ◦ (∇_XE)))^V iv) φJ (X⊗ ˜˜ E)^VY^H = (σp,q−p

2)(X ⊗ (g ◦ (∇YE)))^V +(2σp,q− p

2 )((∇YX)^H− (L_YX)^H

−(˜γ − γ)R(Y, X)) v) φ_{J X˜} HB^V = (p

2− σ_p,q)(∇_XB)^V vi) φJ A˜ ^VB^V = 0

vii) φ_{J A˜} VY^H = 0 viii) φ_{J A˜} V(Y ⊗ ˜E)^V = 0

where R is the curvature tensor of ∇, E be a nowhere zero vector field on M . any vector field X and (1, 1)−tensor fields A,B on M , ˜E = g ◦ E is a covector field on M . The restrictions of ˜J to H + V^E and V^⊥ are endomorphisms, ˜J is a (1, 1)−tensor field on T₁¹(M ).

Proof. i)

φJ X˜ ^HY^H = −(L_Y^HJ )X˜ ^H = −L_Y^HJ X˜ ^H + ˜J L_Y^HX^H

= −L_YH(p

2X^H + (2σ_p,q− p

2 )(X ⊗ ˜E)^V) + ˜J ([Y, X]^H + (˜γ − γ)R(Y, X))

= −p

2[Y, X]^H−p

2(˜γ − γ)R(Y, X)−(2σp,q− p

2 )(∇_Y(X ⊗ ˜E))^V +p

2[Y, X]^H+(2σ_p,q− p

2 )([Y, X]⊗ ˜E)^V+σ_p,q(˜γ − γ)R(Y, X)

= (−p

2+ σp,q)(˜γ − γ)R(Y, X) − (2σ_p,q− p

2 )((∇YX) ⊗ ˜E +X ⊗ (g ◦ (∇_YE)) + (L_YX) ⊗ ˜E)^V

ii)

φJ X˜ ^H(Y ⊗ ˜E)^V = −(L_{(Y ⊗ ˜E)}VJ )X˜ ^H= −L_{(Y ⊗ ˜E)}VJ X˜ ^H+ ˜J L_{(Y ⊗ ˜E)}VX^H

= −L_{(Y ⊗ ˜E)}V(p

2X^H + (2σp,q− p

2 )(X ⊗ ˜E)^V)

− ˜J (∇X(Y ⊗ ˜E))^V

= (p

2−σ_p,q)(Y ⊗ (g ◦ (∇XE)))^V−(2σp,q− p

2 )(∇XY )^H

iii)

φJ (X⊗ ˜˜ E)^V(Y ⊗ ˜E)^V = −(L_{(Y ⊗ ˜E)}VJ )(X ⊗ ˜˜ E)^V

= −L_{(Y ⊗ ˜E)}VJ (X ⊗ ˜˜ E)^V + ˜J L_{(Y ⊗ ˜E)}V(X ⊗ ˜E)^V

= −p

2L_{(Y ⊗ ˜E)}V(X ⊗ ˜E)^V − (2σp,q− p

2 )L_{(Y ⊗ ˜E)}VX^H

= (2σ_p,q− p

2 )((∇_XY ) ⊗ ˜E + Y ⊗ (g ◦ (∇_XE)))^V

iv)

φJ (X⊗ ˜˜ E)^VY^H = −(L_Y^HJ )(X ⊗ ˜˜ E)^V

= −L_Y^HJ (X ⊗ ˜˜ E)^V + ˜J L_Y^H(X ⊗ ˜E)^V

= −L_Y^Hp

2(X ⊗ ˜E)^V+(2σp,q− p

2 )X^H+ ˜J (∇Y(X ⊗ ˜E))^V

= −p

2((∇YX) ⊗ ˜E)^V −p

2(X ⊗ (g ◦ (∇YE)))^V

−(2σp,q− p

2 )([Y, X]^H+ (˜γ − γ)R(Y, X)) +p

2((∇_YX) ⊗ ˜E)^V + (2σp,q− p

2 )(∇_YX)^H +σp,q(X ⊗ (g ◦ (∇YE)))^V

= (σp,q−p

2)(X ⊗ (g ◦ (∇YE)))^V +(2σp,q− p

2 )((∇_YX)^H−(L_YX)^H−(˜γ − γ)R(Y, X)) v)

φ_{J X˜} HB^V = −(L_BVJ )X˜ ^H = −L_BVJ X˜ ^H+ ˜J L_BVX^H

= −L_BV(p

2X^H+ (2σp,q− p

2 )(X ⊗ ˜E)^V) − ˜J (∇XB)^V

= p

2(∇_XB)^V − (2σp,q− p

2 )L_BV(X ⊗ ˜E)^V − σ_p,q(∇_XB)^V

= (p

2 − σ_p,q)(∇XB)^V vi)

φJ A˜ ^VB^V = −(L_BVJ )A˜ ^V = −L_B^VJ A˜ ^V + ˜J L_B^VA^V

= −σ_p,qL_BVA^V

= 0 vii)

φ_{J A˜} VY^H = −(L_YHJ )A˜ ^V = −L_YHJ A˜ ^V + ˜J L_YHA^V

= −L_YHσ_p,qA^V + ˜J (∇_YA)^V

= −σ_p,q(∇_YA)^V + σ_p,q(∇_YA)^V

= 0

viii)

φ_{J A˜} V(Y ⊗ ˜E)^V = −(L_{(Y ⊗ ˜E)}VJ )A˜ ^V = −L_{(Y ⊗ ˜E)}VJ A˜ ^V + ˜J L_{(Y ⊗ ˜E)}VA^V

= −σ_p,qL_{(Y ⊗ ˜E)}VA^V

= 0

2.2. The Vishnevskii Operators applied to vertical and hor- izontal lifts with respect to metallic Riemannian struc- ture.

Definition 2.3. Suppose now that ∇ is a linear connection on M , and let ϕ ∈ =¹₁(M ). We can replace the condition d) of defination 2.1 by

(5) d⁰) ψϕXY = ∇ϕXY − ϕ∇XY

for any X, Y ∈ =¹₀(M ). Then we can consider a new operator by a Vishnevskii operator or ψ_ϕ−operator on M , we shall mean a map ψ_ϕ :

∗

=(M ) → =(M ), which satisfies conditions a), b), c), e) of definition 2.1 and the condition (d⁰) [6, 7, 17].

Let ω ∈ =⁰₁(M ). Using Definition 2.3, we have (ψϕω) (X, Y ) = (ψϕXω)Y

(6)

= (ϕX)(ι_Yω) − X(ι_ϕYω) − ω (∇_ϕXY − ϕ (∇_XY ))

= (∇_ϕXω − ∇_X(ω ◦ ϕ)) Y

for any X, Y ∈ =¹₀(M ),where (ω ◦ ϕ) Y = ω (ϕY ). From (6) we see that ψϕXω = ∇ϕXω − ∇X(ω ◦ ϕ) is a 1−form [17].

Theorem 2.4. ˜J is a metallic structure on T₁¹(M ) defined by (3), ψJ˜

the Vishnevskii operator on M defined by (5), we get the following

results

i) ψJ X˜ ^H(Y ⊗ ˜E)^V = (p

2+ σp,q)(Y ⊗ (g ◦ (∇XE)))^V

−(2σp,q− p

2 )(∇_XY )^H ii) ψ_{J (X⊗ ˜˜ E)}V(Y ⊗ ˜E)^V = (2σ_p,q− p

2 )(∇_XY )⊗ ˜E +Y ⊗(g ◦ (∇_XE)))^V iii) ψJ X˜ ^HY^H = −(2σ_p,q− p

2 )((∇XY ) ⊗ ˜E)^V iv) ψ_{J X˜} HB^V = (p

2− σ_p,q)(∇_XB)^V v) ψJ (X⊗ ˜˜ E)^VY^H = (2σ_p,q− p

2 )(∇XY )^H vi) ψ_{J A˜} V(Y ⊗ ˜E)^V = 0

vii) ψJ A˜ ^VY^H = 0 viii) ψJ A˜ ^VB^V = 0

where R is the curvature tensor of ∇, E be a nowhere zero vector field on M . any vector field X and (1, 1)−tensor fields A,B on M , ˜E = g ◦ E is a covector field on M . The restrictions of ˜J to H + V^E and V^⊥ are endomorphisms, ˜J is a (1, 1)−tensor field on T₁¹(M ).

Proof. i)

ψ_{J X˜} H(Y ⊗ ˜E)^V = ∇^H_{J X˜ H}(Y ⊗ ˜E)^V − ˜J ∇^H_XH(Y ⊗ ˜E)^V

= ∇^H_p

2X^H+(^{2σp,q −p}₂ )(X⊗ ˜E)^V(Y ⊗ ˜E)^V− ˜J ∇^H_XH(Y ⊗ ˜E)^V

= p

2((∇_XY ) ⊗ ˜E)^V +p

2(Y ⊗ (g ◦ (∇_XE)))^V

−p

2((∇_XY ) ⊗ ˜E)^V

−(2σ_p,q− p

2 )(∇_XY )^H + σ_p,q(Y ⊗ (g ◦ (∇_XE)))^V

= (p

2 + σp,q)(Y ⊗ (g ◦ (∇_XE)))^V

−(2σ_p,q− p

2 )(∇_XY )^H

ii)

ψ_{J (X⊗ ˜˜ E)}V(Y ⊗ ˜E)^V = ∇^H_{J (X⊗ ˜˜ E)V}(Y ⊗ ˜E)^V − ˜J ∇^H_{(X⊗ ˜E)V}(Y ⊗ ˜E)^V

= ∇^H_p

2(X⊗ ˜E)^V+(^{2σp,q −p}₂ )X^H(Y ⊗ ˜E)^V

= p 2∇^H

(X⊗ ˜E)^V(Y ⊗ ˜E)^V+(2σp,q− p

2 )∇^H_XH(Y ⊗ ˜E)^V

= (2σ_p,q− p

2 )(∇_X(Y ⊗ ˜E))^V

= (2σ_p,q− p

2 )(∇_XY ) ⊗ ˜E + Y ⊗ (g ◦ (∇_XE)))^V iii)

ψJ X˜ ^HY^H = ∇^H_˜

J X^HY^H − ˜J ∇^H_XHY^H

= ∇^H_p

2X^H+(^{2σp,q −p}₂ )(X⊗ ˜E)^VY^H − ˜J (∇_XY )^H

= p

2(∇XY )^H + (2σp,q− p

2 )∇^H_{(X⊗ ˜E)}VY^H −p

2(∇XY )^H

−(2σp,q− p

2 )((∇_XY ) ⊗ ˜E)^V

= −(2σ_p,q− p

2 )((∇_XY ) ⊗ ˜E)^V iv)

ψJ X˜ ^HB^V = ∇^H_˜

J X^HB^V − ˜J ∇^H_XHB^V

= ∇^H_p

2X^H+(^{2σp,q −p}₂ )(X⊗ ˜E)^VB^V − ˜J (∇_XB)^V

= p

2∇^H_XHB^V + (2σ_p,q− p

2 )∇^H_{(X⊗ ˜E)V}B^V − σ_p,q(∇_XB)^V

= (p

2 − σ_p,q)(∇_XB)^V v)

ψJ (X⊗ ˜˜ E)^VY^H = ∇^H_˜

J (X⊗ ˜E)^VY^H − ˜J ∇^H_{(X⊗ ˜E)}VY^H

= ∇^H_p

2(X⊗ ˜E)^V+(^{2σp,q −p}₂ )X^HY^H

= (2σ_p,q− p

2 )(∇_XY )^H

vi)

ψJ A˜ ^V(Y ⊗ ˜E)^V = ∇^H_˜

J A^V(Y ⊗ ˜E)^V − ˜J ∇^H_AV(Y ⊗ ˜E)^V

= σ_p,q∇^H_AV(Y ⊗ ˜E)^V

= 0 vii)

ψJ A˜ ^VY^H = ∇^H_˜

J A^VY^H − ˜J ∇^H_AVY^H

= ∇^H_σ

p,qA^VY^H

= 0 viii)

ψ_{J A˜} VB^V = ∇^H_˜

J A^VB^V − ˜J ∇^H_AVB^V

= σ_p,q∇^H_AVB^V

= 0

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Ha¸sim C¸ AYIR

Department of Mathematics, Giresun University, 28100, Giresun, Turkey.

E-mail: hasim.cayir@giresun.edu.tr