Metallic Kähler and Nearly Metallic Kahler Manifolds
aa r X i v : . [ m a t h . G M ] J un METALLIC K ¨AHLER AND NEARLY METALLIC K ¨AHLERMANIFOLDS
SIBEL TURANLI, AYDIN GEZER, AND HASAN CAKICIOGLU
Abstract.
In this paper, we construct metallic K¨ahler and nearly metallicK¨ahler structures on Riemanian manifolds. For such manifolds with thesestructures, we study curvature properties. Also we describe linear connectionson the manifold, which preserve the associated fundamental 2-form and satisfysome additional conditions and present some results concerning them.
Primary 53C55; Secondary53C05.
Keywords:
K¨ahler structure, Linear connection, Riemannian curvaturetensor. Basic Definitions and Results
Let M n be an n − dimensional manifold. We point out here and once that allgeometric objects considered in this paper are supposed to be of class C ∞ .The number η = √ ≈ , ... , which is the positive root ofthe equation x − x − p − proportions being a positive root of the equation x p +1 − x p − p =0 , , , , ... ) in [9]. The other called metallic means family or metallic proportionswas introduced by V. W. de Spinadel in [5, 6, 7, 8]. For two positive integers p and q, the positive solution of the equation x − px − q = 0 is named membersof the metallic means family. All the members of the metallic means family arepositive quadratic irrational numbers σ p,q = p + √ p +4 q . These numbers σ p,q are alsocalled ( p, q ) − metallic numbers. Now, we consider the equation x − px + q = 0,where p and q are real numbers satisfying q ≥ −√ q < p < √ q . In thecase, this equation has complex roots as σ cp,q = p ± √ p − q . The complex numbers σ cp,q = p + √ p − q will be called c omplex metallic means family by us. In particular,if p = 1 and q = 1, then the complex metallic means family σ cp,q = p + √ p − q reduces to the complex golden mean: σ c , = √ i , i = − , − tensor field J M which satisfies the relation J M − pJ M + 32 qI = 0 , where I is the identity operator on the Lie algebra of vector fields on M n and p , q are real numbers satisfying q ≥ −√ q < p < √ q . Indeed, an almost complex metallic structure is an example of polynomial structures of degree 2 whichwas generally defined by S. I. Goldberg, K. Yano and N. C. Petridis in ([2] and [3]).Throughout this paper, we will sign by J M an almost complex metallic structure.It is clear that such a structure exists only when M is of even dimension. Becauseof this, we will take n = 2 k .The following result gives relationships between the almost complex structuresand almost complex metallic structures on M k . Proposition 1.1. If J M is an almost complex metallic structure on M k , then J ± = ± (cid:18) σ cp,q − p J M − p σ cp,q − p I (cid:19) are two almost complex structures on M k . Conversely, if J is an almost complexstructure on M k , then J M = p I ± (cid:18) σ cp,q − p (cid:19) J are two almost complex metallic structures on M k , where σ cp,q = p + √ p − q . Proof.
Let us assume that J M is an almost complex metallic structure on M k .Then J = (cid:18) ± (cid:18) σ cp,q − p J M − p σ cp,q − p I (cid:19)(cid:19) = 4 | p − q | J M − p | p − q | J M + p | p − q | I = 1 | p − q | (cid:18) (cid:18) pJ M − qI (cid:19) − pJ M + p I (cid:19) = 1 | p − q | (cid:0) pJ M − qI − pJ M + p I (cid:1) = p − q | p − q | I = − I. In constrast, let J be an almost complex structure on M k . Then J M − pJ M + 32 qI = (cid:18) p I ± (cid:18) σ cp,q − p (cid:19) J (cid:19) − p (cid:18) p I ± (cid:18) σ cp,q − p (cid:19) J (cid:19) + 32 qI = p I ± p p p − q J + (cid:12)(cid:12) p − q (cid:12)(cid:12) J − p I ∓ p p p − q J + 32 I = p I + p − q − p I + 32 I = 0 . (cid:3) Note that the followings satisfy:
ETALLIC K¨AHLER AND NEARLY METALLIC K¨AHLER MANIFOLDS 3 i ) if J is an almost complex structure, then b J = − J is an almost complexstructure, ii ) if J M is an almost complex metallic structure, then c J M = pI − J M is analmost complex metallic structure. In fact c J M − p c J M + 32 qI = ( pI − J M ) − p ( pI − J M ) + 32 qI = p I − pJ M + J M − p I + pJ M + 32 qI = − pJ M + pJ M − qJ + pJ M + 32 qI = 0 . b J and c J M are called the conjugate almost complex structure and the conjugatealmost complex metallic structure, respectively. From Proposition 1.1, it is easy tosee that the almost complex structure J (resp. b J ) defines a J (resp. b J ) − associatedalmost complex metallic structure J M (resp. c J M ), and vice versa. Hence, thereexist an 1 : 1 correspondence between almost complex metallic structures andalmost complex structures on M k .If a manifold M k has an almost complex metallic structure J M , then the pair( M k , J M ) is an almost complex metallic manifold. Recall that a polynomial struc-ture is integrable if the Nijenhuis tensor vanishes [10]. Then, the integrability of J M is equivalent to the vanishing of the Nijenhuis tensor N J M : N J M ( X, Y ) = [ J M X, J M Y ] − J M [ J M X, Y ] − J M [ X, J M Y ] + J M [ X, Y ] . If the almost complex metallic structure J M is integrable, then this structure iscalled a complex metallic structure and the pair ( M k , J M ) is called a complexmetallic manifold. A Riemannian metric on an almost complex metallic manifold( M k , J M ) is hyperbolic with respect to J M if it satisfies(1.1) g ( J M X, Y ) = − g ( X, J M Y )or equivalently(1.2) g ( J M X, J M Y ) = − pg ( X, J M Y ) + 32 qg ( X, Y )for any vector fields X and Y on M k . Also we refer to the conditions (1.1) or(1.2) as the hyperbolic compatibility of g and J M and call g hyperbolic metric . Analmost complex metallic manifold ( M k , J M ) equipped with a hyperbolic metric g is called an almost metallic Hermitian manifold. Proposition 1.2.
Let J (resp. b J ) be an almost complex structure on a Riemannianmanifold ( M k , g ) and J M (resp. c J M ) be a J (resp. b J ) − associated almost complexmetallic structure. The following statements are equivalent:i) g is hyperbolic with respect to J .ii) g is hyperbolic with respect to b J .iii) g is hyperbolic with respect to J M .iv) g is hyperbolic with respect to c J M .Proof. We only prove the equivalence of i) and iv) as the rest of the cases followby the similar argument. SIBEL TURANLI, AYDIN GEZER, AND HASAN CAKICIOGLU
Assuming i) , then, for all vector fields X and Y on M k g (cid:16) c J M X, Y (cid:17) = g (cid:18)(cid:18) p I ± (cid:18) σ cp,q − p (cid:19) b J (cid:19) X, Y (cid:19) = p g ( X, Y ) ± σ cp,q − p g (cid:16) b JX, Y (cid:17) = p g ( X, Y ) ∓ σ cp,q − p g ( JX, Y )= p g ( X, Y ) ± σ cp,q − p g ( X, JY )= − g (cid:18) X, (cid:18) p I ± (cid:18) σ p,q − p (cid:19) b J (cid:19) Y (cid:19) = − g (cid:16) X, c J M Y (cid:17) . Next assuming iv) , then, for all vector fields X and Y on M k g ( JX, Y ) = − g (cid:16) b JX, Y (cid:17) = ∓ g (cid:18)(cid:18) σ p,q − p c J M − p σ p,q − p I (cid:19) X, Y (cid:19) = ∓ σ p,q − p g (cid:16) c J M X, Y (cid:17) ± p σ p,q − p g ( X, Y )= ± σ p,q − p g (cid:16) X, c J M Y (cid:17) ± p σ p,q − p g ( X, Y )= g (cid:18) X, ± (cid:18) σ p,q − p c J M − p σ p,q − p I (cid:19) Y (cid:19) = g (cid:18) X, ± (cid:18) σ p,q − p c J M − p σ p,q − p I (cid:19) Y (cid:19) = g (cid:16) X, b JY (cid:17) = − g ( X, JY ) . (cid:3) From Proposition 1.2, we immediately say that the following statements areequivalent: i) The triple ( M k , g, J ) is an almost Hermitian manifold. ii) The triple ( M k , g, b J ) is an almost Hermitian manifold. iii) The triple ( M k , g, J M ) is an almost metallic Hermitian manifold. iv) The triple ( M k , g, c J M ) is an almost metallic Hermitian manifold.2. Metallic K¨ahler Manifolds
In the following, let ( M k , g, J M ) be an almost metallic Hermitian manifold.Here and in the following, let ∇ always denote the Levi-Civita connection of g . Proposition 2.1.
Let ( M k , g, J M ) be an almost metallic Hermitian manifold and ∇ be the Levi-Civita connection of g . Then the following statements hold:i) ( ∇ X J M ) J M Y = c J M ( ∇ X J M ) Y ii) g (( ∇ X J M ) Y, Z ) = − g ( Y, ( ∇ X J M ) Z ) ETALLIC K¨AHLER AND NEARLY METALLIC K¨AHLER MANIFOLDS 5 for all vector fields
X, Y and Z on M k , where c J M is the conjugate almostcomplex metallic structure.Proof. i) It follows that ∇ X ( J M ) Y = ( ∇ X J M ) J M Y + J M ( ∇ X J M ) Y ∇ X (cid:18) pJ M − qI (cid:19) Y = ( ∇ X J M ) J M Y + J M ( ∇ X J M ) Yp ( ∇ X J M ) Y = ( ∇ X J M ) J M Y + J M ( ∇ X J M ) Y ( ∇ X J M ) J M Y = ( pI − J M ) ( ∇ X J M ) Y ( ∇ X J M ) J M Y = c J M ( ∇ X J M ) Y. ii) The statement is direct consequence of (1.1) and ∇ g = 0. (cid:3) Now, we consider the (0 , − tensor field F , which will later be used for char-acterizing the almost metallic Hermitian manifold. The (0 , − tensor field F isdefined by F ( X, Y, Z ) = g (( ∇ X J M ) Y, Z )for all vector fields
X, Y and Z on M k . Proposition 2.2.
On an almost metallic Hermitian manifold ( M k , g, J M ) , the (0 , − tensor field F satisfies the following properties:i) F ( X, Y, Z ) = − F ( X, Z, Y ) ii) F ( X, J M Y, J M Z ) = qF ( X, Z, Y ) for all vector fields X, Y and Z on M k . Proof. i)
The statement immediately follows from Proposition 2.1. ii)
By means of Proposition 2.1, we have F ( X, J M Y, J M Z ) = g (( ∇ X J M ) J M Y, J M Z )= g ( c J M ( ∇ X J M ) Y, J M Z )= − g ( J M c J M ( ∇ X J M ) Y, Z )= 32 qg (( ∇ X J M ) Z, Y )= 32 qF ( X, Z, Y ) . (cid:3) The 2 − covariant skew-symmetric tensor field ω defined by ω ( X, Y ) = g ( J M X, Y )is the fundamental 2 − form of the almost metallic Hermitian manifold ( M k , g, J M ) . Proposition 2.3.
Let ( M k , g, J M ) be an almost metallic Hermitian manifold and ∇ be the Levi-Civita connection of g . The following statement holds: qF ( X, Y, Z ) + g (cid:16) c J M X, N J M ( Y, Z ) (cid:17) = 3 dω ( X, J M Y, J M Z ) − qdω ( X, Y, Z ) for all vector fields X, Y and Z on M k , where ω is the fundamental − form and N J M is the Nijenhuis tensor of J M .Proof. By the Cartan’s formula, we have(2.1) 3 dω ( X, Y, Z ) = g ( Y, ( ∇ X J M ) Z ) + g ( Z, ( ∇ Y J M ) X ) + g ( X, ( ∇ Z J M ) Y ) . SIBEL TURANLI, AYDIN GEZER, AND HASAN CAKICIOGLU
When writing Y = J M Y and Z = J M Z in (2.1), we find3 dω ( X, J M Y, J M Z ) = g ( J M Y, ( ∇ X J M ) J M Z )(2.2) + g ( J M Z, ( ∇ J M Y J M ) X ) + g ( X, ( ∇ J M Z J M ) J M Y ) . Subtracting (2.2) from (2.1), we have3 dω ( X, J M Y, J M Z ) − q dω ( X, Y, Z )= g ( J M Y, ( ∇ X J M ) J M Z ) + g ( J M Z, ( ∇ J M Y J M ) X )+ g ( X, ( ∇ J M Z J M ) J M Y ) − q g ( Y, ( ∇ X J M ) Z ) − q g ( Z, ( ∇ Y J M ) X ) − q g ( X, ( ∇ Z J M ) Y )= − g (( ∇ X J M ) J M Y, J M Z ) − g (( ∇ J M Y J M ) J M Z, X )+ g ( X, ( ∇ J M Z J M ) J M Y ) + 3 q g (( ∇ X J M ) Y, Z )+ 3 q g (( ∇ Y J M ) Z, X ) − q g ( X, ( ∇ Z J M ) Y )= − g (cid:16) c J M ( ∇ X J M ) Y, J M Z (cid:17) − g (cid:16) c J M ( ∇ J M Y J M ) Z, X (cid:17) + g (cid:16) X, c J M ( ∇ J M Z J M ) Y (cid:17) + 3 q g (( ∇ X J M ) Y, Z ) − g (cid:16) J M ( ∇ Y J M ) Z, c J M X (cid:17) + g (cid:16) c J M X, J M ( ∇ Z J M ) Y (cid:17) = 3 q g (( ∇ X J M ) Y, Z ) + g (cid:16) ( ∇ J M Y J M ) Z, c J M X (cid:17) − g (cid:16) ( ∇ J M Z J M ) Y, c J M X (cid:17) + 3 q g (( ∇ X J M ) Y, Z ) − g (cid:16) J M ( ∇ Y J M ) Z, c J M X (cid:17) + g (cid:16) J M ( ∇ Z J M ) Y, c J M X (cid:17) = 3 qg (( ∇ X J M ) Y, Z ) + g (( ∇ J M Y J M ) Z − ( ∇ J M Z J M ) Y + J M ( ∇ Z J M ) Y − J M ( ∇ Y J M ) Z, c J M X )= 3 qF ( X, Y, Z ) + g (cid:16) c J M X, N J M ( Y, Z ) (cid:17) . Thus, we have our relation. (cid:3)
Theorem 2.4.
Let ( M k , g, J M ) be an almost matallic Hermitian manifold and ∇ be the Levi-Civita connection of g . The conditions dω = 0 and N J M = 0 areequivalent to ∇ J M = 0 . Proof.
It easy to see that ( ∇ X ω )( Y, Z ) = g (( ∇ X J M ) Y, Z ) = F ( X, Y, Z ) for anyvector fields
X, Y, Z on M k . Assuming that F ( X, Y, Z ) = 0, i.e., ∇ J M = 0. Then dω = 0 obviously. Furthermore, by Proposition 2.3, we obtain N J M = 0.Conversely, assuming that dω = 0 and N J M = 0. The result immediately followsfrom by Proposition 2.3. (cid:3) If the fundamental 2 − form ω is closed, i.e., dω = 0, then we will call thetriple ( M k , g, J M ) an almost metallic K¨ahler manifold. Moreover, if dω = 0 and N J M = 0, we will call the triple ( M k , g, J M ) a metallic K¨ahler manifold. In view ETALLIC K¨AHLER AND NEARLY METALLIC K¨AHLER MANIFOLDS 7 of Theorem 2.4, an almost metallic Hermitian manifold ( M k , g, J M ) is a metallicK¨ahler manifold if and only if ∇ J M = 0.2.1. Curvature properties.
Let ( M k , g, J M ) be a metallic K¨ahler manifold. De-note by R and S the Riemannian curvature tensor and the Ricci tensor of M k ,respectively. Theorem 2.5.
Let ( M k , g, J M ) be a metallic K¨ahler manifold. The followingstatements hold:i) R ( X, Y ) J M Z = J M R ( X, Y ) Z and R ( J M X, J M Y ) Z = − pR ( J M X, Y ) Z + q R ( X, Y ) Z for all vector fields X, Y, Z on M k . ii) S ( J M X, J M Y ) = (cid:16) p − q p + q (cid:17) S ( X, Y ) + (cid:16) pq − q p (cid:17) S ( X, J M Y ) and (cid:0) q (cid:1) S ( X, Y ) − p S ( X, J M Y ) = − q trace c J M R ( X, J M Y ) for all vectorfields X, Y on M k .Proof. i) By applying the Ricci identity to J M , the first relation immediately followsfrom ∇ J M = 0. For any vector fields X, Y, Z and W on M k , we get g ( R ( J M X, J M Y ) Z, W )= R ( J M X, J M Y, Z, W ) = R ( Z, W, J M X, J M Y )= R ( W, Z, J M Y, J M X ) = g ( R ( W, Z ) J M Y, J M X )= g ( J M R ( W, Z ) Y, J M X ) = − pR ( W, Z, Y, J M X ) + 3 q R ( W, Z, Y, X )= − pR ( J M X, Y, Z, W ) + 3 q R ( X, Y, Z, W )= − pg ( R ( J M X, Y ) Z, W ) + 3 q g ( R ( X, Y ) Z, W )from which we have R ( J M X, J M Y ) Z = − pR ( J M X, Y ) Z + 3 q R ( X, Y ) Z. ii) Let { e , e , ..., e k } be an orthonormal basis of M k . For any vector fields X, Y on M k , we have S ( J M X, J M Y )(2.3) = X g ( R ( e i , J M X ) J M Y, e i )= X g ( R ( J M e i , J M X ) J M Y, J M e i )= X g ( J M R ( J M e i , J M X ) Y, J M e i )= − X g (cid:0) R ( J M e i , J M X ) Y, J M e i (cid:1) = − P X g ( R ( J M e i , J M X ) Y, J M e i ) + 3 q X g ( R ( J M e i , J M X ) Y, e i )= − p X R ( J M e i , J M X, Y, J M e i ) + 3 q X R ( J M e i , J M X, Y, e i )= p X R ( J M e i , X, Y, J M e i ) − pq X R ( e i , X, Y, J M e i ) − pq X R ( J M e i , X, Y, e i ) + 9 q X R ( e i , X, Y, e i ) . SIBEL TURANLI, AYDIN GEZER, AND HASAN CAKICIOGLU
Also we yield − pq X R ( e i , X, Y, J M e i )(2.4) = − pq X g ( R ( e i , X ) Y, J M e i )= 3 pq X g ( J M R ( e i , X ) Y, e i )= 3 pq X g ( R ( e i , X ) J M Y, e i )and − pq X R ( J M e i , X, Y, e i )(2.5) = − pq X R (cid:16) J M c J M e i , X, Y, c J M e i (cid:17) = − pq X R (cid:18) q e i , X, Y, ( pI − J M ) e i (cid:19) = − p q X R ( e i , X, Y, e i ) + 9 pq X R ( e i , X, Y, J M e i )= − p q X g ( R ( e i , X ) Y, e i ) + 9 pq X g ( R ( e i , X ) Y, J M e i )= − p q X g ( R ( e i , X ) Y, e i ) − pq X g ( J M R ( e i , X ) Y, e i )= − p q X g ( R ( e i , X ) Y, e i ) − pq X g ( R ( e i , X ) J M Y, e i ) . Substituting (2.4) and (2.5) into (2.3), we get S ( J M X, J M Y )= p X R ( J M e i , X, Y, J M e i ) + 3 pq X g ( R ( e i , X ) J M Y, e i ) − p q X g ( R ( e i , X ) Y, e i ) − pq X g ( R ( e i , X ) J M Y, e i )+ 9 q X R ( e i , X, Y, e i )= p X g ( R ( J M e i , X ) Y, J M e i ) + 3 pq X g ( R ( e i , X ) J M Y, e i ) − p q X g ( R ( e i , X ) Y, e i ) − pq X g ( R ( e i , X ) J M Y, e i )+ 9 q X g ( R ( e i , X ) Y, e i )= p S ( X, Y ) + 3 pq S ( X, J M Y ) − p q S ( X, Y ) − pq S ( X, J M Y ) + 9 q S ( X, Y )= (cid:18) p − p q q (cid:19) S ( X, Y ) + (cid:18) pq − pq (cid:19) S ( X, J M Y ) . Thus, we completes the proof of the first formula of ii).
ETALLIC K¨AHLER AND NEARLY METALLIC K¨AHLER MANIFOLDS 9
With the help of the first Bianchi’s identity, we have S ( X, Y )= X g ( R ( e i , X ) Y, e i )= 23 q X g (cid:16) c J M J M R ( e i , X ) Y, e i (cid:17) = 23 q X g (cid:16) c J M R ( e i , X ) J M Y, e i (cid:17) = − q X g (cid:16) c J M R ( X, J M Y ) e i , e i (cid:17) − q X g (cid:16) c J M R ( J M Y, e i ) X, e i (cid:17) = − q X g (cid:16) c J M R ( X, J M Y ) e i , e i (cid:17) − q X g (cid:16) c J M R ( J M Y, J M e i ) X, J M e i (cid:17) = − q X g (cid:16) c J M R ( X, J M Y ) e i , e i (cid:17) + 23 q X g (cid:16) c J M J M R ( J M Y, J M e i ) X, e i (cid:17) = − q X g (cid:16) c J M R ( X, J M Y ) e i , e i (cid:17) + X g ( R ( J M Y, J M e i ) X, e i )= − q X g (cid:16) c J M R ( X, J M Y ) e i , e i (cid:17) − p X g ( R ( J M Y, e i ) X, e i )+ 3 q X g ( R ( Y, e i ) X, e i )= − q T race c J M R ( X, J M Y ) − pS ( X, J M Y ) + 3 q S ( X, J M Y )which completes the proof. (cid:3) Theorem 2.6.
Let ( M k , g, J M ) be a metallic K¨ahler manifold. The Ricci tensor S of M k satisfies (cid:18) q (cid:19) ( ∇ Z S ) ( X, Y ) − P ( ∇ Z S ) ( X, J M Y )= (cid:18) q (cid:19) ( ∇ X S ) ( Z, Y ) − P ( ∇ X S ) ( Z, J M Y )+ (cid:18) q + 1 (cid:19) ( ∇ J M Y S ) (cid:16) X, c J M Z (cid:17) − P ( ∇ J M Y S ) (cid:16) X, c J M Z (cid:17) for all vector fields X, Y, Z on M k .Proof. From the second relation of ii) in Theorem 2.5 and the second Bianchi’sidentity we have (cid:18) q (cid:19) ( ∇ Z S ) ( X, Y ) − P ( ∇ Z S ) ( X, J M Y )(2.6) = − q X g (cid:16) c J M ( ∇ Z R ) ( X, J M Y ) e i , e i (cid:17) = − q X g (cid:16) c J M ( ∇ X R ) ( Z, J M Y ) e i , e i (cid:17) − q X g (cid:16) c J M ( ∇ J M Y R ) ( X, Z ) e i , e i (cid:17) = (cid:18) q (cid:19) ( ∇ X S ) ( Z, Y ) − P ( ∇ X S ) ( Z, J M Y ) − q X g (cid:16) c J M ( ∇ J M Y R ) ( X, Z ) e i , e i (cid:17) . When writing Z = J M Z ve Y = c J M Y in the second relation of ii) in Theorem 2.5,we find (cid:18) q (cid:19) ( ∇ J M Z S ) (cid:16) X, c J M Y (cid:17) − P ( ∇ J M Z S ) (cid:16) X, J M c J M Y (cid:17) = − q X g (cid:16) c J M ( ∇ J M Z R ) (cid:16) X, J M c J M Y (cid:17) e i , e i (cid:17)(cid:18) q (cid:19) ( ∇ J M Z S ) (cid:16) X, c J M Y (cid:17) − pq ∇ J M Z S ) ( X, Y )= − X g (cid:16) c J M ( ∇ J M Z R ) ( X, Y ) e i , e i (cid:17) from which it follows that − q X g (cid:16) c J M ( ∇ J M Y R ) ( X, Z ) e i , e i (cid:17) = (cid:18) q + 1 (cid:19) ( ∇ J M Y S ) (cid:16) X, c J M Z (cid:17) − p ( ∇ J M Y S ) ( X, Z ) . Substituting the last relation into (2.6), the result follows. (cid:3) Nearly metallic K¨ahler Manifolds
Let ( M k , g, J M ) be an almost metallic Hermitian manifold. Following terminolo-gies used in [11] for the almost Hermitian manifolds, we can say that for a givenalmost metallic Hermitian manifold ( M k , g, J M ), if the the fundamental 2 − form ω satisfies the following relation:(3.1) ( ∇ X ω )( Y, Z ) + ( ∇ Y ω )( X, Z ) = 0for all vector fields
X, Y and Z , then we will call the triple ( M k , g, J M ) a nearlymetallic K¨ahler manifold. It is clear that the relation (3.1) is equivalent to(3.2) ( ∇ X J M ) Y + ( ∇ Y J M ) X = 0 . Next we will prove the following two propositions.
Proposition 3.1.
On a nearly metallic K¨ahler manifold ( M k , g, J M ) , the (0 , − tensorfield F satisfies the following properties:i) F ( J M X, Y, J M Z ) = q F ( Y, X, Z ) ii) F ( J M X, J M Y, Z ) = − pF ( Y, X, c J M Z ) + q F ( Y, X, Z ) for all vector fields X, Y and Z on M k . Proof. i)
It follows that F ( J M X, Y, J M Z ) = g (( ∇ J M X J M ) Y, J M Z )= − g (( ∇ Y J M ) J M X, J M Z )= − g (cid:16) c J M ( ∇ Y J M ) X, J M Z (cid:17) = g ( J M c J M ( ∇ Y J M ) X, Z )= 3 q g (( ∇ Y J M ) X, Z )= 3 q F ( Y, X, Z ) ETALLIC K¨AHLER AND NEARLY METALLIC K¨AHLER MANIFOLDS 11 ii)
We calculate F ( J M X, J M Y, Z ) = g (( ∇ J M X J M ) J M Y, Z )= g (cid:16) c J M ( ∇ J M X J M ) Y, Z (cid:17) = g (cid:16) c J M ( ∇ Y J M ) X, c J M Z (cid:17) = − pg (( ∇ Y J M ) X, c J M Z ) + 3 q g (( ∇ Y J M ) X, Z )= − pF ( Y, X, c J M Z ) + 3 q F ( Y, X, Z ) . (cid:3) Theorem 3.2.
A nearly metallic K¨ahler manifold is integrable if and only if it isa metallic K¨ahler manifold.Proof.
On a nearly metallic K¨ahler manifold ( M k , g, J M ), the Nijenhuis tensor of J M verifies N J M ( X, Y ) = [ J M X, J M Y ] − J M [ J M X, Y ] − J M [ X, J M Y ] + J M [ X, Y ]= ( ∇ J M X J M ) Y − ( ∇ J M Y J M ) X − J M ( ∇ X J M ) Y + J M ( ∇ Y J M ) X = − ( ∇ Y J M ) J M X + ( ∇ X J M ) J M Y − J M ( ∇ X J M ) Y − J M ( ∇ X J M ) Y = − c J M ( ∇ Y J M ) X + c J M ( ∇ X J M ) Y − J M ( ∇ X J M ) Y = 2 c J M ( ∇ X J M ) Y − J M ( ∇ X J M ) Y = 2( pI − J M ) ( ∇ X J M ) Y from which we say that N J M = 0 if and only if ∇ J M = 0. This expression completesthe proof. (cid:3) Curvature properties.
Coordinate systems in a nearly metallic K¨ahler man-ifold ( M k , g, J M ) are denoted by ( U, x i ), where U is the coordinate neighbourhoodand x i , i = 1 , , ..., k are the coordinate functions. Substituting X = ∂∂x i and Y = ∂∂x j in (3.1) and (3.2), one respectively has ∇ i ω jm + ∇ j ω im = 0and ∇ i ( J M ) hj + ∇ j ( J M ) hi = 0 . Contraction with respect to i and h in the last relation, we get ∇ i ( J M ) ij = 0. Theorem 3.3.
The Ricci and Ricci* curvature tensors in a nearly metallic K¨ahlermanifold ( M k , g, J M ) satisfy S jt ( J M ) ti = − q S ∗ jt ( c J M ) ti if and only if ∇ m ∇ j ω im = 0 , where ω im are the components of the fundamental − form ω .Proof. When applied the Ricci identity to ( J M ) hi , one has ∇ k ∇ j ( J M ) hi − ∇ j ∇ k ( J M ) hi = R hkjt ( J M ) ti − R tkji ( J M ) ht , where R hkjt are components of the Riemannian curvature tensor R . Contractionthe above relation with respect to k and h gives ∇ h ∇ j ( J M ) hi − ∇ j ∇ h ( J M ) hi = R hhjt ( J M ) ti − R thji ( J M ) ht ∇ h ∇ j ( J M ) hi = S jt ( J M ) ti − R thji ( J M ) ht (3.3) = S jt ( J M ) ti − R hjil g lt ( J M ) ht = S jt ( J M ) ti − R hjil ω hl = S jt ( J M ) ti − H ji . Here S jt are the components of the Ricci curvature tensor and ω hl are the con-travariant components of the fundamental 2 − form ω . Also note that the tensor H ji is anti-symmetric. In fact H ji = R hjil ω hl = 12 ( R hjil + R hjil ) ω hl = 12 ( R hjil − R ljih ) ω hl and similarly H ij = R hijl ω hl = 12 ( R hijl + R hijl ) ω hl = 12 ( R hijl − R lijh ) ω hl The sum of the above relations gives H ij + H ji = 12 ( R hjil − R ljih + R hijl − R lijh ) ω hl = 0 . The tensor S ∗ given by [11] S ∗ ji = − H jt ( J M ) ti is called the Ricci* curvature tensor of M k . It is easy to see that(3.4) S ∗ jt ( c J M ) ti = − qH ji . From (3.3) and (3.4) we obtain ∇ t ∇ j ( J M ) ti = S jt ( J M ) ti + 23 q S ∗ jt ( c J M ) ti ∇ t ∇ j ( g mt ω im ) = S jt ( J M ) ti + 23 q S ∗ jt ( c J M ) ti g mt ∇ t ∇ j ω im = S jt ( J M ) ti + 23 q S ∗ jt ( c J M ) ti ∇ m ∇ j ω im = S jt ( J M ) ti + 23 q S ∗ jt ( c J M ) ti which finishes the proof. (cid:3) Theorem 3.4.
In a nearly metallic K¨ahler manifold ( M k , g, J M ) , the Ricci tensor S is hyperbolic with respect to the almost complex metallic structure J M .Proof. Since the tensor H is an anti-symmetric, we have H ij + H ji = S it ( J M ) tj + S jt ( J M ) ti − (cid:16) ∇ h ∇ i ( J M ) hj + ∇ h ∇ j ( J M ) hi (cid:17) S it J M tj + S jt J M ti − ∇ h (cid:16) ∇ i ( J M ) hj + ∇ j ( J M ) hi (cid:17) S ti ( J M ) tj = − S jt ( J M ) ti . (cid:3) ETALLIC K¨AHLER AND NEARLY METALLIC K¨AHLER MANIFOLDS 13
Theorem 3.5.
In a nearly metallic K¨ahler manifold ( M k , g, J M ) , the Ricci* ten-sor S ∗ is hyperbolic with respect to the conjugate almost complex metallic structure c J M .Proof. For the Ricci* curvature tensor S ∗ in a nearly metallic K¨ahler manifold( M k , g, J M ), with the help of ω lh = − ω hl and the properties of Riemannian cur-vature tensor, we have 23 q S ∗ jm ( c J M ) mi = − H ji q S ∗ jm ( c J M ) mi = − R hjil ω lh q S ∗ jm ( c J M ) mi = −
12 ( R hjil + R hjil ) ω lh (3.5) 23 q S ∗ jm ( c J M ) mi = −
12 ( R hjil − R ljih ) ω lh and similarly 23 q S ∗ im ( c J M ) mj = − H ij q S ∗ im ( c J M ) mj = − R hijl ω lh (3.6) 23 q S ∗ im ( c J M ) mj = −
12 ( R hjil − R lijh ) ω lh The sum of (3.5) and (3.6) gives23 q (cid:16) S ∗ jm ( c J M ) mi + S ∗ im ( c J M ) mj (cid:17) = −
12 ( R hjil − R ljih + R hjil − R lijh ) ω lh S ∗ jm ( c J M ) mi + S ∗ im ( c J M ) mj = 0 S ∗ jm ( c J M ) mi = − S ∗ im ( c J M ) mj . Since S ∗ im is symmetric, consequently S ∗ jm ( c J M ) mi = − S ∗ mi ( c J M ) mj . (cid:3) Theorem 3.6.
In a nearly metallic K¨ahler manifold ( M k , g, J M ) , the relationshipbetween the scalar and scalar* curvature is as follows: S ∗ c = 32 qS c + pS jt ω jt − k∇ J M k , where ω jt are the covariant components of the fundamental − form ω .Proof. In a nearly metallic K¨ahler manifold ( M k , g, J M ), transvecting ∇ j ω im = −∇ j ω mi = ∇ m ω ji with ω ji , it follows that( ∇ j ω im ) ω ji = 0 . Taking covariant derivative ∇ k of the last relation, we find ∇ k { ( ∇ j ω im ) ω ji } = 0 (cid:0) ∇ k ∇ j ω im (cid:1) ω ji + ( ∇ j ω im ) (cid:0) ∇ k ω ji (cid:1) = 0 (3.7) ( ∇ k ∇ m ω ji ) ω ji + ( ∇ m ω ji ) (cid:0) ∇ k ω ji (cid:1) = 0Transvecting (3.7) by g km , we find g km ( ∇ k ∇ m ω ji ) ω ji + g km ( ∇ m ω ji ) (cid:0) ∇ k ω ji (cid:1) = 0( ∇ m ∇ m ω ji ) ω ji + g km (cid:16) ∇ m g jt ( J M ) ti (cid:17) (cid:16) ∇ k g is ( J M ) js (cid:17) = 0( ∇ m ∇ m ω ji ) ω ji + g km g jt g is (cid:16) ∇ m ( J M ) ti (cid:17) (cid:16) ∇ k ( J M ) js (cid:17) = 0 (cid:18) S jt ( J M ) ti + 23 q S ∗ jt ( c J M ) ti (cid:19) ω ji + k∇ J M k = 0 − (cid:18) S jt ( J M ) ti + 23 q S ∗ jt ( c J M ) ti (cid:19) ω ij + k∇ J M k = 0 − (cid:18) S jt ( J M ) ti + 23 q S ∗ jt ( c J M ) ti (cid:19) ( J M ) in g nj + k∇ J M k = 0 S jt (cid:16) ( J M ) ti ( J M ) in (cid:17) + 23 q S ∗ jt (cid:16) ( c J M ) ti ( J M ) in (cid:17) g nj + k∇ J M k = 0 S jt (cid:18) p ( J M ) tn − qδ tn (cid:19) + 23 q S ∗ jt (cid:18) qδ tn (cid:19) g nj + k∇ J M k = 0 (cid:18) pS jt (cid:0) J M tn (cid:1) − qS jn + S ∗ jn (cid:19) g nj + k∇ J M k = 0 pS jt ( J M ) tn g nj − qS jn g nj + S ∗ jn g nj + k∇ J M k = 0 pS jt ω tj − qS c + S ∗ c + k∇ J M k = 0 − pS jt ω jt − qS c + S ∗ c + k∇ J M k = 0 S ∗ c = 32 qS c + pS jt ω jt − k∇ J M k . (cid:3) Linear connections
In this section, by employing the method proposed in [4] for anti-Hermitian man-ifolds we search for linear connections with torsion on an almost metallic Hermitianmanifold ( M k , g, J M ). We will be calling these connections linear connections ofthe first type and of the second type, respectively.Following the method from [4], we have the following definition. Definition 4.1.
A linear connection e ∇ X Y = ∇ X Y + S ( X, Y ) on an almostmetallic Hermitian manifold ( M k , g, J M ) satisfying e ∇ ω = 0 and S J M ( X, Y, Z ) + S J M ( X, Z, Y ) = 0 is called a linear connection of the first type , where S is a(1 , − tensor field, ω is the fundamental 2 − form and S J M ( X, Y, Z ) = g ( S ( X, Y ) , J M Z ). ETALLIC K¨AHLER AND NEARLY METALLIC K¨AHLER MANIFOLDS 15
For the covariant derivative of the fundamental 2 − form ω with respect to e ∇ , wefind ( e ∇ X ω )( Y, Z ) = e ∇ X ( ω ( Y, Z )) − ω ( e ∇ X Y, Z ) − ω ( Y, e ∇ X Z )(4.1) = ∇ X ( ω ( Y, Z )) − ω ( ∇ X Y + S ( X, Y ) , Z ) − ω ( Y, ∇ X Z + S ( X, Z ))= ∇ X ( ω ( Y, Z )) − ω ( ∇ X Y, Z ) − ω ( Y, ∇ X Z ) − ω ( S ( X, Y ) , Z ) − ω ( Y, S ( X, Z ))= ( ∇ X ω )( Y, Z ) − ω ( S ( X, Y ) , Z ) − ω ( Y, S ( X, Z ))= ( ∇ X ω )( Y, Z ) − g ( J M S ( X, Y ) , Z ) − g ( J M Y, S ( X, Z ))= ( ∇ X ω )( Y, Z ) + g ( S ( X, Y ) , J M Z ) − g ( S ( X, Z ) , J M Y )= ( ∇ X ω )( Y, Z ) + S J M ( X, Y, Z ) − S J M ( X, Z, Y )for any vector fields
X, Y, Z on M k . In view of the assumptions for e ∇ , from (4.1)we get S J M ( X, Y, Z ) = −
12 ( ∇ X ω )( Y, Z ) g ( S ( X, Y ) , J M Z ) = − g (( ∇ X J M ) Y, Z ) g ( J M S ( X, Y ) , Z ) = 12 g (( ∇ X J M ) Y, Z ) J M S ( X, Y ) = 12 ( ∇ X J M ) YS ( X, Y ) = 13 q c J M ( ∇ X J M ) Y, i.e., the linear connection of the first type is given by e ∇ = ∇ + q c J M ( ∇ J M ). Wecalculate( e ∇ X g )( Y, Z ) = X ( g ( Y, Z )) − g ( e ∇ X Y, Z ) − g ( Y, e ∇ X Z )= X ( g ( Y, Z )) − g ( ∇ X Y + 13 q c J M ( ∇ X J M ) Y, Z ) − g ( Y, ∇ X Z + 13 c J M ( ∇ X J M ) Z )= ( ∇ X g )( Y, Z ) − q g ( c J M ( ∇ X J M ) Y, Z ) − q g ( Y, c J M ( ∇ X J M ) Z )= − q g (( ∇ X J M ) J M Y, Z ) + 13 q g ( c J M Y, ( ∇ X J M ) Z )= 13 q g ( J M Y, ( ∇ X J M ) Z ) + 13 q g ( c J M Y, ( ∇ X J M ) Z )= p q g ( Y, ( ∇ X J M ) Z ) . Hence, we get the following result.
Theorem 4.2.
On an almost metallic Hermitian manifold ( M k , g, J M ) , the linearconnection of the first type is given by e ∇ = ∇ + 13 q c J M ( ∇ J M ) and it is metric with respect to g if and only if the almost metallic Hermitian man-ifold ( M k , g, J M ) is a metallic K¨ahler manifold. In the case, the linear connectionof the first type and the Levi-Civita connection coincides each other. Definition 4.3.
A linear connection e ∇ X Y = ∇ X Y + S ( X, Y ) on an almostmetallic Hermitian manifold ( M k , g, J M ) satisfying e ∇ ω = 0 and S J M ( X, Y, Z ) + S J M ( Z, Y, X ) = 0 is called a linear connection of the second type.We can write( ∇ X ω )( Y, Z ) + S J M ( X, Y, Z ) − S J M ( X, Z, Y ) = 0( ∇ Y ω ) ( Z, X ) + S J M ( Y, Z, X ) − S J M ( Y, X, Z ) = 0( ∇ Z ω ) ( X, Y ) + S J M ( Z, X, Y ) − S J M ( Z, Y, X ) = 0from which, by virtue of S J M ( X, Y, Z ) + S J M ( Z, Y, X ) = 0, it follows that2 S J M ( X, Y, Z ) = ( ∇ X ω ) ( Y, Z ) + ( ∇ Y ω ) ( Z, X ) + ( ∇ Z ω ) ( X, Y )(4.2) 2 g ( S ( X, Y ) , J M Z ) = dω ( X, Y, Z ) − g ( J M S ( X, Y ) , Z ) = dω ( X, Y, Z ) . On an almost metallic K¨ahler manifold we get S = 0, which means that e ∇ = ∇ .Hence, we have: Theorem 4.4.
If an almost metallic Hermitian manifold ( M k , g, J M ) is almostmetallic K¨ahler, the linear connection of the second type is egual to ∇ . If the almost metallic Hermitian manifold ( M k , g, J M ) is nearly metallic K¨ahler,then (4.2) reduces to − g ( J M S ( X, Y ) , Z ) = 3 ( ∇ X ω ) ( Y, Z ) g ( J M S ( X, Y ) , Z ) = − g (( ∇ X J M ) Y, Z ) J M S ( X, Y ) = −
32 ( ∇ X J M ) YS ( X, Y ) = − q c J M ( ∇ X J M ) Y. Thus, we get:
Theorem 4.5.
If an almost metallic Hermitian manifold ( M k , g, J M ) is nearlymetallic K¨ahler, the linear connection of the second type is given by e ∇ = ∇ − q c J M ( ∇ J M ) . References [1] M. Crasmareanu, C. Hretcanu, Golden differential geometry. Chaos Solitons Fractals (5)(2008), 1229–1238.[2] S. I. Goldberg, K. Yano, Polynomial structures on manifolds. Kodai Math. Sem. Rep. (1970), 199-218. ETALLIC K¨AHLER AND NEARLY METALLIC K¨AHLER MANIFOLDS 17 [3] S. I. Goldberg and N. C. Petridis, Differentiable solutions of algebraic equations on manifolds.Kodai Math. Sem. Rep. (1973), 111–128.[4] A. Salimov, On anti-Hermitian metric connections. C. R. Math. Acad. Sci. Paris (9)(2014), 731–735.[5] V. W. de Spinadel, The metallic means family and multifractal spectra. Nonlinear Anal. Ser.B: Real World Appl. (6) (1999), 721-745.[6] V. W. de Spinadel, The family of metallic means. Vis. Math. (3)(2002), 279-288.[8] V. W. de Spinadel, The metallic means family and renormalization group techniques. Proc.Steklov Inst. Math., Control in Dynamic Systems, suppl. (2000), 194-209.[9] Stakhov AP. Introduction into algorithmic measurement theory. Moscow 1977 [in Russian].[10] J. Vanzura, Integrability conditions for polynomial structures. Kodai Math. Sem. Rep. (1-2) 1976, 42-50.[11] K. Yano, Differential geometry on complex and almost complex spaces. International seriesof monographs in pure and applied mathematics, vol. 49, Pergamon Press, The Macmillan,New York, 1965. Erzurum Technical University, Faculty of Science, Department of Mathematics,Erzurum-TURKEY.
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