We first introduce Golden maps between Golden Riemannian manifolds, give an example and show that such map is harmonic. Then we investigate the constancy of certain maps from Golden Riemannian manifolds to various manifolds by imposing the holomorphic-like map condition. Then we consider the reverse case and show that all such
maps are constant

Bayram Şahin

Mehmet Akif Akyol

English

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MATHEMATICAL COMMUNICATIONS 333Math. Commun. 19(2014), 333–342Golden maps between Golden Riemannian manifolds andconstancy of certain mapsBayram Şahin1,∗and Mehmet Akif Akyol21 Department of Mathematics, Faculty of Arts and Sciences, İnönü University, TR-44 280Malatya, Turkey2 Department of Mathematics, Faculty of Arts and Sciences, Bingöl University,TR-12 000 Bingöl, TurkeyReceived January 24, 2014; accepted July 18, 2014Abstract. We first introduce Golden maps between Golden Riemannian manifolds, givean example and show that such map is harmonic. Then we investigate the constancy ofcertain maps from Golden Riemannian manifolds to various manifolds by imposing theholomorphic-like map condition. Then we consider the reverse case and show that all suchmaps are constant.AMS subject classifications: 53C15, 53C25, 53D15Key words: Golden Riemannian manifold, almost contact metric manifold, almost prod-uct manifold, holomorphic map1. IntroductionManifolds equipped with certain differential-geometric structures possess rich geo-metric structures and such manifolds and maps between them have been studiedwidely in differential geometry. Indeed, almost complex manifolds, almost contactmanifolds and almost product manifolds and maps between such manifolds havebeen studied extensively by many authors. Such manifolds are defined by a (1, 1)-tensor field Φ such that the square of Φ satisfies certain conditions, like Φ2 = −I,Φ2 = I or Φ2 = −I + η ⊗ ξ, where η and ξ are 1-form and a vector field.The number ϕ = (1 +√5)/2 = 1, 618... which is a solution of the equationx2 − x − 1 = 0, represents the golden ratio. Geometrically, Golden ratio impliesthat if a unit segment is divided into two subsegments, then both the ratio of theentire segment to the major subsegment and the ratio of the major subsegmentto the minor subsegment must equal ϕ. The Golden ratio has been used in manydifferent areas, particularly, in arts and architecture. Being inspired by the Goldenratio, the notion of Golden manifold M was defined in [4] by a tensor field Φ on Msatisfying Φ2 = Φ+ I. The authors studied properties of Golden manifolds and theyshowed that Φ is an automorphism of the tangent bundle TM and its eigenvaluesare ϕ and 1− ϕ. They also defined Golden Riemannian manifolds and investigated∗Corresponding author. Email addresses: bayram.sahin@inonu.edu.tr (B. Şahin),makyol@bingol.edu.tr (M.A.Akyol)http://www.mathos.hr/mc c©2014 Department of Mathematics, University of Osijek334 B. Şahin and M.A.Akyoltheir submanifolds in [5]. Moreover, the integrability of Golden structures has beeninvestigated in [8].To compare two manifolds or to study the geometry of a manifold M by usinganother manifold N , it is useful to define a map F from M to N by imposing certainconditions on F . In this manner, holomorphic maps between two complex manifoldshave nice geometric properties. For instance, any holomorphic map between Kaehlermanifolds is harmonic [6]. By adapting the notion of holomophic maps, similar mapsdefined between almost contact manifolds [9] and other manifolds endowed with ageometric structure.In this paper, we study a new map between Golden Riemannian manifolds byimposing a holomorphic-like condition for the first time as far as we know. Weshow that such map is a harmonic map, and then we obtain certain conditions forsuch maps to be totally geodesic. We also provide a simple elementary example.Moreover, we also check the existence of such maps between Golden Riemannianmanifolds and another manifold equipped with a differentiable structure (almostcomplex, almost contact, almost product, almost para-contact) and surprisingly wefind that there are no non-constant such maps.2. PreliminariesIn this section, we give a brief information for almost complex manifolds, almostcontact metric manifolds, almost product manifolds, almost para-contact metricmanifolds, Golden Riemannian manifolds. We note that throughout this paper allmanifolds and bundles, along with sections and connections, are assumed to be ofclass C∞. A map is always a C∞ map between manifolds.2.1. Almost complex manifoldsLet M ′ be a 2n− dimensional real manifold. An almost complex structure J on M ′is a tensor field J : TM ′ → TM ′ such thatJ2 = −I, (1)where I is the identity transformation. Then (M ′, J) is called an almost complexmanifold [13].A smooth map ϕ : M ′1 → M ′2 between almost complex manifolds (M ′1, J1) and(M ′2, J2) is called an almost complex (or holomorphic) map if dϕ (J1X) = J2dϕ (X)for X ∈ Γ (TM ′1) , where J1 and J2 are complex structures of M ′1 and M ′2, respec-tively.2.2. Almost contact metric manifoldsAn n−dimensional differentiable manifold M is said to have an almost contact struc-ture (φ, ξ, η) if it carries a tensor field φ of type (1, 1), a vector field ξ and 1− formη on M , respectively, such thatφ2 = −I + η ⊗ ξ, φξ = 0, η ◦ φ = 0, η (ξ) = 1, (2)Golden maps and constancy of certain maps 335where I is the identity transformation. The almost contact structure is said to benormal if N + 2dη ⊗ ξ = 0, where N is the Nijenhuis tensor of φ. Suppose that aRiemannian metric tensor g is given in M and satisfies the conditiong (φX, φY ) = g (X,Y )− η (X) η (Y ) , η (X) = g (X, ξ) . (3)Then (M,φ, ξ, η, g) is called an almost contact metric manifold [3].2.3. Almost product manifoldsLet N be an n−dimensional manifold with a tensor of type (1, 1) such thatF 2 = I, (4)where I is the identity transformation. Then we say that N is an almost productmanifold with almost product structure F. We putQ =12(I + F ) , Q′=12(I − F ) . (5)Then we haveQ+Q′= I, Q2 = Q, Q′2 = Q′, QQ′= Q′Q = 0 (6)andF = Q−Q′. (7)If an almost product manifold N admits a Riemannian metric g such thatg (FX,FY ) = g (X,Y )for any vector fields X and Y on N, then N is called an almost product Riemannianmanifold [13].2.4. Almost para-contact metric manifoldsAn n−dimensional differentiable manifold N ′ is said to have an almost para-contactstructure (φ′, ξ′, η′) if it carries a tensor field φ′ of type (1, 1) , a vector field ξ′ anda 1−form η′ on N ′, respectively, such thatφ′2 = I − η′ ⊗ ξ′, η′ (ξ′) = 1, φ′ξ′ = 0, η′ ◦ φ′ = 0 (8)where I is the identity tranformation. Suppose that a Riemannian metric tensor g′is given in N ′ and satisfies the conditiong′ (φ′X,φ′Y ) = g′ (X,Y )− η′ (X) η′ (Y ) (9)g′ (X, ξ) = η′ (X) , g′ (X,φ′Y ) = −g′ (φ′X,Y ) (10)for any vector fields X and Y on N ′, then (N ′, φ′, ξ′, η′, g′) is called an almost para-contact metric manifold [11, 14].336 B. Şahin and M.A.Akyol2.5. Golden Riemannian manifoldsLet(M̄, g)be a Riemannian manifold. A golden structure on(M̄, g)is an (1, 1)tensor field P which satisfies the equationP 2 = P + I, (11)where I is the identity transformation. We say that the metric g is P compatible ifg (PX, Y ) = g (X,PY ) (12)for all X,Y ∈ Γ(TM̄). If we substitute PX into X in (12), equation (12) may alsowritten asg (PX,PY ) = g(P 2X,Y)= g ((P + I)X,Y ) = g (PX, Y ) + g (X,Y ) .The Riemannian metric (12) is called P−compatible and(M̄, P, g)is named aGolden Riemannian manifold [4]. It is known ([4]) that a Golden structure P isintegrable if the Nijenhuis tensor NP vanishes. In [8], the authors show that aGolden structure is integrable if and only if ∇P = 0, where ∇ is the Levi-Civitaconnection of g.2.6. Harmonic mapsLet (M, gM ) and (N, gN) be Riemannian manifolds and suppose that P : M → Nis a smooth mapping between them. Then the differential dP of P can be viewedas a section of the bundle Hom(TM,P−1TN)→ M, where P−1TN is the pull-back bundle which has fibres(P−1TN)p= TP (p)N, p ∈ M. Hom(TM,P−1TN)has a connection ∇ induced from the Levi-Civita connection ∇M and the pullbackconnection. Then the second fundamental form of P is given by∇dP (X,Y ) = ∇PXdP (Y )− dP(∇MX Y)(13)for X,Y ∈ Γ (TM) . It is known that the second fundamental form is symmetric. Asmooth map P : (M, gM ) → (N, gN ) is said to be harmonic if trace ∇dP = 0. Thetension field of P is the section τ (P ) of Γ(P−1TN)defined byτ (P ) = divdP =m∑i=1∇dP (ei, ei) , (14)where {e1, ..., em} is a local orthonormal frame on M. Then it follows that P isharmonic if and only if τ (P ) = 0. For more information, see [1].3. Golden maps between Golden manifoldsIn this section, we give a new notion, namely a Golden map, and show that suchmap is harmonic. We also investigate conditions for a Golden map to be totallygeodesic.Golden maps and constancy of certain maps 337Definition 1. Let ϕ be a smooth map from a Golden Riemannian manifold (M,P, g)to a Golden Riemannian manifold (N,P ′, g′). Then ϕ is called a Golden map if thefollowing condition is satisfied.dϕP = P ′dϕ . (15)We provide the following elementary example.Example 1. Let ϕ : R4 → R2 be a map defined byϕ (x1, x2, x3, x4) =(x1 + x24,x3 + x44).Then, by direct calculationsker dϕ = span{X1 =∂∂x1− ∂∂x2, X2 =∂∂x3− ∂∂x4}and(kerdϕ )⊥ = span{Z1 =∂∂x1+∂∂x2, Z2 =∂∂x3+∂∂x4}.Then considering Golden structures on R4 and R2 defined byΦ (a1, a2, a3, a4) = (φa1, φa2, (1− φ) a3, (1− φ) a4)andΦ′ (a1, a2) = (φa1, (1− φ) a2) ,where φ and 1 − φ are eigenvalues of Golden structures [4]. It is easy to see thatdϕ (PZ1) = P′dϕ (Z1) and dϕ (PZ2) = P′dϕ (Z2) . Thus ϕ is a Golden map.From now on, when we mention a Golden Riemannian manifold, we will assumethat its almost Golden structure is integrable.Lemma 1. Let ϕ be a Golden map from a Golden Riemannian manifold (M,P, g)to a Golden Riemannian manifold (N,P ′, g′) such that dϕP = P ′dϕ is satisfied.Then we have(∇dϕ ) (X,PY ) = (∇dϕ ) (PX, Y ) (16)for X,Y ∈ Γ (TM) .Proof. For X,Y ∈ Γ (TM) , from (13) and (11) we have(∇dϕ ) (X,PY ) = ∇ϕXdϕ(P 2Y − Y)− dϕ(∇MX PY).Then using (15) we get(∇dϕ ) (X,PY ) = ∇ϕXP ′2dϕ (Y )−∇ϕXdϕ (Y )− dϕ(∇MX PY).Since P ′ is integrable, from (11) we obtain(∇dϕ ) (X,PY ) = P ′∇ϕXdϕ (Y )− dϕ(∇MX PY).338 B. Şahin and M.A.AkyolIntegrable P and (15) also imply(∇dϕ ) (X,PY ) = P ′(∇ϕXdϕ (Y )− dϕ(∇MX Y)).Thus using (13) we arrive at(∇dϕ ) (X,PY ) = P ′ (∇dϕ ) (X,Y ) . (17)Since ∇dϕ is symmetric, we obtain (16).The following theorem shows that a Golden map is a harmonic map.Theorem 1. Every Golden map from a Golden Riemannian manifold (M,P, g) toa Golden Riemannian manifold (N,P ′, g′) is harmonic.Proof. First, from (16) and (17), we find(∇dϕ ) (PX,PY ) = P ′2 (∇dϕ ) (X,Y ) . (18)Let {e1, e2, ..., en} be a basis of TpM, p ∈ M. Since P is an isomorphism, from [12]we know that {Pe1, P e2, ..., P en} is also a basis of TpM, p ∈ M. Then from (11) and(18), we haven∑i=1(∇dϕ ) (Pei, P ei) =n∑i=1P ′ (∇dϕ) (ei, ei) +n∑i=1(∇dϕ ) (ei, ei)τ (ϕ) = P ′τ (ϕ) + τ (ϕ) ,which givesP ′τ (ϕ) = 0.Since P ′ is an isomorphism on N , we getτ (ϕ) = 0,which completes the proof.Remark 1. We note that for any C2 real valued function f defined on an open subsetof a Riemannian manifold M , the equation △f = 0 is called Laplace’s equation andsolutions are called harmonic functions on U . Let F : M −→ N be a smooth mapbetween Riemannian manifolds. Then F is called a harmonic morphism if, for everyharmonic function f : V −→ R defined an open subset V of N with F−1(V ) non-empty, the composition f ◦F is harmonic on F−1(V ). A smooth map F : M −→ Nbetween Riemannian manifolds is a harmonic morphism if and only if F is bothharmonic and horizontally weakly conformal [7] and [10]. In this respect, a Goldenmap is a good candidate for a harmonic morphism.We now give a necessary and sufficient condition for a map ϕ to be totallygeodesic. We recall that a map ϕ is totally geodesic if ∇dϕ = 0. A geometricinterpretation of a totally geodesic map is that it maps every geodesic in the totalmanifold into a geodesic in the base manifold in proportion to arc lengths.Golden maps and constancy of certain maps 339Theorem 2. Let ϕ be Golden map from a Golden Riemannian manifold (M,P, g)to a Golden Riemannian manifold (N,P ′, g′) . Then ϕ is totally geodesic if and onlyif(∇dϕ ) (PX,PY ) = P ′ (∇dϕ ) (X,Y ) , (19)for X,Y ∈ Γ (TM) .Proof. It is obvious from (18).4. Constancy of some maps from Golden Riemannian mani-foldsIn this section we investigate constancy of certain maps from Golden Riemannianmanifolds or to Golden manifolds by imposing holomorphic-like conditions. We firstcheck the situation for a map between Golden Riemannian manifolds and almostcomplex manifolds.Theorem 3. Let ϕ be a smooth map from a Golden Riemannian manifold(M̄, P, g)to an almost complex manifold (M ′, J) such that the condition dϕP = Jdϕ is satis-fied. Then ϕ is a constant map.Proof. Let(M̄, P, g)be a Golden Riemannian manifold and (M ′, J) an almostcomplex manifold. Suppose that ϕ : M̄ → M ′ satisfiesdϕ (PX) = Jdϕ (X) , X ∈ Γ(TM̄). (20)Then apply J to the above equation and using (1) and (11), we getdϕ (PX) + dϕ (X) = −dϕ (X) , X ∈ Γ(TM̄). (21)Applying J to (21) again and using (20), we havedϕ (PX) + dϕ (X) + Jdϕ (X) = −Jdϕ (X) , X ∈ Γ(TM̄). (22)Then (20) implies that−3dϕ (PX) = dϕ (X) , X ∈ Γ(TM̄). (23)From (21) and (23) we obtaindϕ (X) = 0,which shows that ϕ is constant.In a similar way, we have the following result.Theorem 4. Let ϕ be a smooth map from an almost complex manifold (M ′, J) toa Golden Riemannian manifold(M̄, P, g)such that the condition dϕJ = Pdϕ issatisfied. Then ϕ is a constant map.340 B. Şahin and M.A.AkyolRemark 2. Besides holomorphic maps, anti-holomorphic maps have also been stud-ied by many authors under the condition J ′dF = −dFJ , where F is a map be-tween almost complex manifolds and J, J ′ are almost complex structures, respec-tively. By following similar computations above, one can find that the notion ofanti-holomorphic maps does not work for Golden manifolds.The following result shows that a smooth map satisfying a compatible conditionbetween Golden Riemannian manifolds and almost contact metric manifolds is alsoconstant.Theorem 5. Let ϕ be a smooth map from a Golden Riemannian manifold(M̄, P, g)to an almost contact metric manifold (M,φ, ξ, η, g) such that the condition dϕP =φdϕ is satisfied. Then ϕ is a constant map.Proof. Let(M̄, P, g)be a Golden Riemannian manifold and (M,φ, ξ, η, g) an almostcontact metric manifold. Suppose that ϕ : M̄ → M satisfiesdϕ (PX) = φdϕ (X) , X ∈ Γ(TM̄). (24)Then apply φ to the above equation and using (2) and (11), we getdϕ (PX) = −2dϕ (X) + η (dϕ (X)) ξ, X ∈ Γ(TM̄). (25)Then applying φ to (25) again and using (24) and (2), we have−3dϕ (PX) = dϕ (X) . (26)From (25) and (26), we obtain−5dϕ (PX) = η (dϕ (X)) ξ, X ∈ Γ(TM̄). (27)Again applying φ to (27), we getφdϕ (X) = 0, X ∈ Γ(TM̄). (28)Using (28) in (27) we conclude thatη (dϕ (X)) = 0, X ∈ Γ(TM̄). (29)Then applying ϕ to (29) we get−dϕ (X) + η (dϕ (X)) ξ = 0which givesdϕ (X) = 0.This completes the proof.In a similar way, we have the following result.Theorem 6. Let ϕ be a smooth map from an almost contact metric manifold(M,φ, ξ, η, g) to a Golden Riemannian manifold(M̄, P, g)such that the conditiondϕφ = Pdϕ is satisfied. Then ϕ is a constant map.Golden maps and constancy of certain maps 341We now check a similar situation for a map between Golden Riemannian mani-folds and almost product manifolds.Theorem 7. Let ϕ be a smooth map from a Golden Riemannian manifold(M̄, P, g)to an almost product manifold (N,F ) such that the condition dϕP = Fdϕ is satisfied.Then ϕ is a constant map.Proof. Let(M̄, P, g)be a Golden Riemannian manifold and (N,F ) an almost prod-uct manifold. Suppose that ϕ : M̄ → N satisfiesdϕ (PX) = Fdϕ (X) , X ∈ Γ(TM̄). (30)Then apply F to the above equation and using (4) and (11), we getdϕ (PX) = 0, X ∈ Γ(TM̄). (31)Applying F to (31) again and using (30), we havedϕ (PX) = −dϕ(X), X ∈ Γ(TM̄). (32)From (32) and (31) we obtaindϕ (X) = 0,which shows that ϕ is constant.In a similar way, we have the following result.Theorem 8. Let ϕ be a smooth map from an almost product manifold (N,F ) toa Golden Riemannian manifold(M̄, P, g)such that the condition dϕF = Pdϕ issatisfied. Then ϕ is a constant map.Finally, we check the same problem for almost para-contact metric manifolds.Theorem 9. Let ϕ be a smooth map from a Golden Riemannian manifold(M̄, P, g)to an almost para-contact metric manifold (N ′, φ′, ξ′, η′, g′) such that the conditiondϕP = φ′dϕ is satisfied. Then ϕ is a constant map.Proof. Let(M̄, P, g)be a Golden Riemannian manifold and (N ′, φ′, ξ′, η′, g′) analmost para-contact metric manifold. Suppose that ϕ : M̄ → N ′ satisfiesdϕ (PX) = φ′dϕ (X) , X ∈ Γ(TM̄). (33)Then apply φ′ to the above equation and using (8) and (11), we getdϕ (PX) + dϕ (X) = dϕ (X)− η′ (dϕ (X)) ξ′, X ∈ Γ(TM̄). (34)Then applying φ′ to (34) again and using (33) and (8), we havedϕ (PX) = −dϕ (X) . (35)From (34) and (35), we obtaindϕ (X) = η′ (dϕ (X)) ξ′, X ∈ Γ(TM̄). (36)Again applying φ′ to (36), we getφ′dϕ (X) = 0, X ∈ Γ(TM̄). (37)From (37) we have dϕ (X) = 0, which shows that ϕ is constant.342 B. Şahin and M.A.AkyolIn a similar way, we have the following result.Theorem 10. Let ϕ be a smooth map from an almost para-contact metric manifold(N ′, φ′, ξ′, η′, g′) to a Golden Riemannian manifold(M̄, P, g)such that the conditiondϕφ′ = Pdϕ is satisfied. Then ϕ is a constant map.References[1] P.Baird, J. C.Wood, Harmonic Morphisms between Riemannian Manifolds, Lon-don Mathematical Society Monographs, No. 29, Oxford University Press, Oxford,2003.[2] D.E. Blair, Contact Manifolds in Riemannian Geometry, Lectures Notes in Mathe-matics, Springer-Verlag, New York, 1976.[3] D.E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Birkhäuser,Boston, 2002.[4] M.Crasmareanu, C.E.Hretçanu, Golden differential geometry, Chaos, Solitons &Fractals 38(2008), 1229–1238.[5] M.Crasmareanu, C.E. Hretçanu, Applications of the Golden ratio on Riemannianmanifolds, Turkish J. Math. 33(2009), 179–191.[6] J. Eells, H. J. 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Golden maps between Golden Riemannian manifolds and constancy of certain maps

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