Applications of Φ-Operators to Hypercomplex Geometry - Springer Link

Adv. Appl. Clifford Algebras 22 (2012), 185–201 © 2011 Springer Basel AG 0188-7009/030185-17 published online March 30, 2011 DOI 10.1007/s00006-011-0291-7

Advances in Applied Clifford Algebras

Applications of Φ-Operators to Hypercomplex Geometry∗ Arif Salimov† and Seher Aslanci Abstract. The main aim of this paper is to study relations between Φoperators, hyperholomorphic functions and the theory of lifts in tensor bundles. Model of lifts of affinor fields was found by using of Φ-operators. After that, hyperholomorphic properties of pure cross-sections were investigated by the help of this model. Keywords. Hyperholomorphic function, pure tensor field, tensor bundle, complete lift.

1. Introduction 1.1. Am -modules of regular representations Let Am be an associative commutative unitial algebra (hypercomplex algebra) of order m over field of real numbers R. We consider the exact (monomorphic) representation Φ : Am → End Ln of algebra Am in a linear space Ln over R. Note that the algebra Am admits in its vector space the so-called regular representation, given by linear operators Sa (x) = ax, where a is a fixed element of Am . It is not difficult to see that the regular representation is exact. For the regular representation we have β aσ , α, β, σ = 1, . . . , m, (Sa )βα = Cσα β are structure constants of the algebra Am . In particular, to the where Cσα β ). It is known, base units eσ ∈ Am there correspond the matrices Sσ = (Cσα that for the linear operator (affinor) to belong to regular representation {Sσ } the necessary and sufficient condition is that it commutes with all Sσ [2]. With the aid of regular representation we build the so-called r-regular representation of algebra Am in linear space Ln (n = mr), which is also exact, ∗ This

paper is supported by The Scientific and Technological Research Council of Turkey (TBAG-108T590) † Corresponding author.

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and the matrix of r-regular representation has the form (ϕ) σ

i j

= δvu (Sσ )α β,

where δvu is the Kronecker symbol and u, v = 1, . . . , r; i, j = 1, . . . , n. In this work we consider only r-regular representations of algebra Am . Let Lr (Am ) be an A-module or a module over algebra Am of order r [11, p.65], which is defined with the aid of the operators {ϕ} or r-regular representation Φ : Am → End Ln , where {ϕ} = Φ(Am ) ⊂ End Ln , n = mr. Note that the A-module Lr (Am ) arises after comparison ∗

ξ i = ξ (u−1)m+α = ξ uα → ξ u = ξ uα eα . In fact, if

η i = ϕ ij ξ j , where σ

ϕ

σ

i j

∈ Φ(Am ), then

α vβ η uα = δvu Cσβ ξ ,

or

∗

∗

α uβ η u = η uα eα = Cσβ ξ eα = eσ eβ ξ uβ = eσ ξ u . ∗

The vector transformation law for quantities ξ u is verified after the definition of fundamental group of module Lr (Am ). The fundamental group of the module Lr (Am ) is realized in Ln as the subgroup Gϕ ⊂ GL(n, R), which preserves affinors of representation, i.e. ∀P ∈ Gϕ and ∀ϕ ∈ Φ(Am ), ϕP = P ϕ, Det(Pji ) = 0 Thus, any block of matrix P of order m commutes with all Sσ . From this fact, we have Cα , Pji = Δσu u σα are arbitrary coefficients subject to regularity condition Det(Pji ) where Δσu u ∗

∗



e , i.e. the vector = 0. It is easily seen, that ξ u = Auu ξ u , where Auu = Δσu u σ module Lr (Am ) over algebra Am is defined correctly. 1.2. Hyperholomorphic functions Let z = xα eα be a variable in algebra Am and f 1 (x), f 2 (x), . . . , f m (x) be the set of functions of all xα . Then ω = f α (x)eα is a function of z. We define the differentials dω = df α eα , dz = dxα eα . The function ω = ω(z) is called hyperholomorphic if there exists a function ω  (z) = εβ (∂β f α )eα ( eβ are components of the unit ε ∈ Am ) such that dω = ω  (z)dz.

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The necessary and sufficient condition for hyperholomorphity of function ω = ω(z) is the condition [2] Sσ D = DSσ , γ (Cσβ ),

(1)

α ( ∂f ). ∂xβ

where Sσ = D= Condition (1) will be called the Scheffers condition [9] . In particular, in case of the algebra of complex numbers A2 = R(i), i2 = −1, the Scheffers condition coincides with Cauchy-Riemann conditions. If we consider the algebra of dual numbers A2 = R(ε), ε2 = 0, then from (1) it follows that the condition of existence of derivative dω ω  (z) = , ω = f 1 (x1 , x2 ) + εf 2 (x1 , x2 ), z = x1 + εx2 , dz has the form ∂f 1 ∂f 2 ∂f 1 = 0, = . ∂x2 ∂x2 ∂x1 Here from we obtain that the dualholomorphic function ω = ω(z) has the structure (2) ω = F (x1 ) + ε(x2 F  (x1 ) + G(x1 )). The dualholomorphic function in form (2) is called synectic. In particular, if in (2) G(x1 ) = 0, then the dualholomorphic function in form (2) is called the natural extension of real differentiable function F (x1 ) to the algebra R(ε). The notion of hyperholomorphic function of several variables from algebra is introduced in a natural way [11, p. 94]: The function ω = f α (x1 , . . . , xrm )eα is hyperholomorphic of z u = (u−1)m+α x eα , u, v = 1, . . . , r, if and only if the Scheffers condition is valid for Jacobian matrix D(f 1 , . . . , f m ) , u = 1, . . . , r. D(x(u−1)m+1 , . . . , xum ) 1.3. Hypercomplex structures Let Mn be a connected manifold of class C ∞ . The field of endomorphisms Π = {ϕ} is called an algebric hypercomplex Π- structure over Mn . By the structure we mean affinors ϕ, α = 1, . . . , m, which correspond to the base α

units eα ∈ Am under the isomorphism Φ. Then γ i ϕim ϕm j = Cαβ ϕj . α

β

(3)

γ

If Φ is the r-regular representation of algebra Am , then the hypercomplex Π-structure is called r-regular Π-structure over Mn (n = mr). Note that if A2 is complex algebra, then the r-regularity of its representation over Mn at once follows (3). Therefore an almost complex structure over M2r is an example of r-regular Π-structure. A.P. Shirokov proved in [1] that in the tangent bundle the r-regular Π-structure arises in natural way and is defined by algebra of dual numbers.

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If the coordinate neighbourhood U ⊂ Mn is endowed with affine connection in which ϕ = 0, ∀ϕ ∈ Π, then such a connection is called a Πconnection. Π-structure is called integrable, if Mn admits a smooth atlas of local charts such that any affinor ϕ ∈ Π in any of the charts of this atlas has constant components. Π-structure is called almost integrable, if in a neighbourhood of any point of Mn there exists at least one Π-connection without torsion. It is known, that any integrable r-regular Π-structure is almost integrable and vice versa. From the facts mentioned above it follows that if on Mrm the r-regular Π-structure is given, then tangent space Tx (Mrm ) at any point x ∈ Mn is transformed to the module Lr (Am ) over algebra Am . Moreover, if the rregular Π-structure on Mn is integrable, then as it is proved in [2] the adapted charts on Mrm consists of charts that are connected by hyperholomorphic transition functions, i.e. Mn carries the structure of hyperholomorphic manifold of order r over algebra Am : Xr (A).

2. Φϕ -Operator and Hyperholomorphic Tensor Fields We shall now derive explicit expressions for Φϕ -operator which applied to an arbitrary pure tensor field of type (p, q). Explicit formulae of Φϕ -operator for pure tensor fields of types (1, q) and (o, q) are given in [9]. Also in [9] derives relations between the geometry of hyperholomorphic B-manifolds (Norden manifolds) and Φϕ -operator. We note that, Φϕ -operator is extension of the operator of Lie derivation LX , X ∈ 10 (Mn ) to affinor fields. The main purpose of the present paper is to discuss relations between these operators, hyperholomorphic tensors and the theory of lifts on pure cross-sections in tensor bundles of types (p, q).   ∗

Let Am be a Frobenius hypercomplex algebra and K =

∗ u1 ...up

K v1 ...vq

be a hypercomplex tensor field on Xr (Am ). Then the real model of such a i ...i tensor field is a tensor field K = Kj11...jpq on Mmr of the same order that is independent of whether its vector or covector arguments is subject to the action of affinors ϕ, α = 1, . . . , m . Such tensor fields are said to be pure α

with respect to Π = {ϕ}, α = 1, . . . , m. They were studied by many authors α

([2], [4]-[9], [11]). Applied to K ∈ pq (Mn ), p + q > 1, the purity means that for any X1 , X2 , . . . , Xq ∈ 10 (Mn ) and ξ1 , ξ2 , . . . , ξp ∈ 01 (Mn ) the following conditions should hold: K(ϕX1 , X2 , . . . , Xq , ξ1 , ξ2 , . . . , ξp ) = K(X1 , ϕX2 , . . . , Xq , ξ1 , ξ2 , . . . , ξp ) = . . . = K(X1 , X2 , . . . , ϕXq , ξ1 , ξ2 , . . . , ξp ) = K(X1 , X2 , . . . , Xq , ϕ ξ1 , ξ2 , . . . , ξp ) = K(X1 , X2 , . . . , Xq , ξ1 , ϕ ξ2 , . . . , ξp ) = . . .

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= K(X1 , X2 , . . . , Xq , ξ1 , ξ2 , . . . , ϕ ξp ), where ϕ is the adjoint operator of ϕ. The vector (covector) field and scalar is considered to be pure, by convention. ∗

We denote by pq (Mn ) the modul of all pure tensor fields of type (p, q) on Mn with respect to the affinor field ϕ ∈ 11 (Mn ). We now fix a positive integer λ. If K and L are pure tensor fields of types (p1 , q1 ) and (p2 , q2 ) C

respectively, then the tensor product of K and L with contraction: K ⊗ L = i1 ...mλ ...ip r ...rp 1 Ls11 ...mλ2 ...sq2 is also pure tensor field. We shall prove only the Kj1 ...jq 1

∗

∗

case when K ∈ 11 (Mn ) and L ∈ 02 (Mn ). In fact, we have C

C

(K ⊗ L)(ϕX, Y ) = K(L(ϕX, Y )) = K(L(X, ϕY )) = (K ⊗ L)(X, ϕY ). ∗

We shall now make the direct sum (Mn ) =

∞ 

∗

pq (Mn ) into an alge-

p,q=0 C

bra over the real number R by defining the pure product (denoting by ⊗) of ∗

∗

K ∈ pq11 (Mn ) and L ∈ pq22 (Mn ) as follows: ⎧ i1 ...mλ ...ip r1 ...rp 1 ⎪ Kj1 ...jq Ls1 ...mλ2 ...sq2 , for λ ≤ p1 , q2 ⎪ 1 ⎪ ⎪ ⎪ (λ − fixed positive integer) ⎪ ⎪ ⎨ i1 ...ip1 r1 ...mμ ...rp2 C C K L , for μ ≤ p2 , q1 s ...s 1 q2 j1 ...mμ ...jq1 ⊗ : (K, L) → K ⊗ L = ⎪ ⎪ (μ − fixed positive integer) ⎪ ⎪ ⎪ ⎪ = 0, p = 0 0, for p 1 2 ⎪ ⎩ 0, for q1 = 0, q2 = 0. Let K ∈ 10 (Mn ) and L ∈ Λq2 (Mn ) be a q2 -form. Then the pure product coincides with interior product iX L. ∗

Definition 1. A map Φϕ : (Mn ) → (Mn ) ((Mn ) =

∞ 

pq (Mn )) is a

p,q=0

Φϕ -operator on Mn if (a) Φϕ is linear with respect to constant coefficients, ∗

(b) for all p, q Φϕ : pq (Mn ) → pq+1 (Mn ), ∗

(c) for all K, L ∈ (Mr ), C

C

C

Φϕ (K ⊗ L) = (Φϕ K) ⊗ L + K ⊗ Φϕ L (d) for all X, Y ∈ 10 (Mn ), ΦϕX Y = −(LY ϕ)X, where LY is the Lie derivation with respect to Y . (e) for all ω ∈ 01 (Mn ) and X, Y ∈ 10 (Mn ), ΦϕX (iY ω) = (ϕX)(iY ω) − X(iϕY ω), where iY ω = (ωY ) ∈ 00 (Mn ). Remark 1 . It follows that Φϕ possesses also the following property: ΦϕX (ω(Y1 , . . . , Yq )) = (ϕX)(ω(Y1 , . . . , Yq )) − X(ω(ϕY1 , . . . , Yq )).

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Proof. We shall prove the formula for the case q = 2. By (d) of Definition and purity of ω, we have ΦϕX (ω(Y, Z)) = ΦϕX ((iY ω)Z)) = ΦϕX (iZ (iY ω)) = (ϕX)(iZ (iY ω)) − X(iϕZ (iY ω)) = (ϕX)(iY ω)Z − X(iY ω)(ϕZ) = (ϕX)(ω(Y, Z)) − X(ω(ϕY, Z)).



∗

Let K ∈ 1q (Mn ). Using the condition (c) of Definition, we have, for any operator Φϕ : ΦϕX (K(Y1 , . . . , Yq )) = (ΦϕX K)(Y1 , . . . , Yq ) +

q 

K(Y1 , . . . , ΦϕX Yλ , . . . , Yq ).

λ=1

Then (d) of Definition implies (Φϕ K)(X; Y1 , . . . , Yq ) = (ΦϕX K)(Y1 , . . . , Yq ) = −(LK(Y1 ,...,Yq ) ϕ)X +

q 

K(Y1 , . . . , (LYλ ϕ)X, . . . , Yq ).

λ=1

Remark 2. If K = ϕ, then Φϕ ϕ(X, Y ) = Nϕ (X, Y ), where Nϕ is the Nijenhuis tensor of ϕ. ∗

Using (e) by similar devices for ω ∈ 0q (Mn ), we have (Φϕ ω)(X; Y1 , . . . , Yq ) = (LϕX ω − LX (ω ◦ ϕ))(Y1 , Y2 , . . . , Yq ) + −

q  λ=2 q 

ω(Y1 , Y2 , . . . , ϕ(LX Yλ ), . . . , Yq ) ω(ϕY1 , Y2 , . . . , LX Yλ , . . . , Yq ),

λ=2 ∗

Let t ∈ pq (Mn ), p > 1, q ≥ 1. We now define a pure tensor field of type ∗

(0, q) tξ1 ,...,ξp ∈ 0q (Mn ) by tξ1 ,...,ξp (Y1 , . . . , Yq ) = t(Y1 , . . . , Yq , ξ 1 , . . . , ξ p ), where tξ1 ,...,ξp has the components of the form: i ...i

1

2

p

(tξ1 ,...,ξp )j1 j2 ...jq = tj11 ...jpq ξi1 ξi2 . . . ξip . According to Remark 1, we find ΦϕX t(Y1 , . . . , Yq , ξ 1 , . . . , ξ p ) = ΦϕX tξ1 ,...,ξp (Y1 , . . . , Yq ) = (ϕX)tξ1 ,...,ξp (Y1 , . . . , Yq ) − Xtξ1 ,...,ξp (ϕY1 , . . . , Yq ) = (ϕX)t(Y1 , . . . , Yq , ξ 1 , . . . , ξ p ) − Xt(ϕY1 , . . . , Yq , ξ 1 , . . . , ξ p ).

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∗

Then for Φϕ t for t ∈ pq (Mn ), p > 1, q ≥ 1, is, by definition, a tensor field of type (p, q + 1) given by Φϕ t(X; Y1 , . . . , Yq , ξ 1 , . . . , ξ p ) = (ΦϕX t)(Y1 , . . . , Yq , ξ 1 , . . . , ξ p ) = (ϕX)t(Y1 , . . . , Yq , ξ 1 , . . . , ξ p ) − Xt(ϕY1 , Y2 . . . , Yq , ξ 1 , . . . , ξ p ) + −

q  λ=1 p 

(4)

t(Y1 , . . . , (LYλ ϕ)X, . . . , Yq , ξ 1 , . . . , ξ p ) t(Y1 , . . . , Yq , ξ 1 , . . . ,

μ=1

(LϕX ξ μ − LX (ξ μ ◦ ϕ)), . . . , ξ p ) The case where p > 1, q ≥ 1 the following theorem is true (for p = 0, q ≥ 1 see [9]). Theorem 1. Let on Mrm be given the integrable Frobenius r-regular hyper∗

complex Π-structure. For hypercomplex tensor field t of type (p, q) on Xr (A) to be A-hyperholomorphic tensor field it is necessary and sufficient that ∗

Φϕ t = 0, α = 1, . . . , m , t ∈ (Mrm ) . α

Proof. By setting X = ∂k , Yλ = ∂jλ , ξ m = dxim , λ = 1, . . . , q, m = 1, . . . , p i ...ip in the equation of (4), we see that the components (Φϕ t)kj1 1 ...j of Φϕ t with q α

α

respect to local coordinate system x1 , . . . , xrm may be expressed as follows:  q   i ...ip i1 ...ip i1 ...ip i ...ip m m (Φϕ t)kj1 1 ...j = ϕ ∂ t − ∂ (t(ϕ )) + ϕ ∂ tj11 ...m....j k jλ k m j1 ...jq k j1 ...jq q q α

α

α

+

p  μ=1

i

λ=1

i ...m...ip

(∂k ϕimμ − ∂m ϕkμ )tj11 ...jq

α

.

By virtue of [2] (see also [11, p.72]) i ...i

σ u1 ...up

α1 ε tj11 ...jpq =  v1 ...vq Cσε C β1 λ1 λ

λ

λ

q−2 q−1 q Cβλ21λ2 . . . Cβq−1 λq−1 Cβq λq Cβq+1 λq+1 . . .

λ

σ u1 ...up

α ...α

q+p−2 βq+1 α2 Cβq+p−1 . . . ϕβq+p αp =  v1 ...vq Bσβ1 1 ...βpq βq+p ϕ

(ϕβα -Frobenius metric), in the adapted charts we have (iμ = uμ αμ , jλ = vλ βλ , k = wγ, λ = 1, . . . , q, μ = 1, . . . , p): i ...i

i ...i

i ...i

p 1 p 1 p (Φϕ t)kj1 1 ...j = ϕm k ∂m tj1 ...jq − ∂k (t(ϕ))j1 ...jq q α

α

α

α ...α

μ ...up λ λ ...up σ = (Cαγ ∂wμ vu11...v − Cασ ∂wγ vu11...v )Bαβ1 1 ...βpq = 0 q q

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or μ ...up λ λ ...up σ Cαγ ∂wμ vu11...v = Cασ ∂wγ vu11...v q q ∗ u ...u 1 p v1 ...vq

which is the Scheffers condition of A-hyperholomorphity of t σ u1 ...up

=

v1 ...vq eσ with respect to local coordinates z u = xuα eα from Xr (A). This completes the proof.  be  a manifold with non-integrable algebraic hypercomRemark 3. Let Mn  plex structure Π = ϕ , i.e. (Mn , Π) is an almost algebraic hypercomplex α

manifold. In this case, when Φϕ t = 0, t is said to be almost hyperholomorphic. α

3. Lifts of the Vector Fields on a Cross-Section in the Tensor Bundle Let Mn be a differentiablemanifold of class C ∞ and finite dimension n. Tqp (P ) is, by definition, the tensor bundle of Then the set Tqp (Mn ) = P ∈Mn

type (p, q) over Mn , where



denotes the disjoint union of the tensor spaces for all P ∈ Mn . For any point P˜ of Tqp (Mn ) such that P˜ ∈ Tqp (Mn ), the surjective correspondence P˜ → P determines the natural projection π : Tqp (Mn ) → Mn . The projection π defines the natural differentiable manifold structure of Tqp (Mn ), that is, Tqp (Mn ) is a C ∞ -manifold of dimension n+np+q . If xj are local coordinates in a neighborhood U of P ∈ Mn , then a tensor t i ...i at P which is an element of Tqp (Mn ) is expressible in the form (xj , tj11 ...jpq ),

Tqp (P )

i ...i

where tj11 ...jpq are components of t with respect to natural base. We may i ...i ¯ consider (xj , t 1 p ) = (xj , xj ) = xJ , j = 1, . . . , n, ¯j = n + 1, . . . , n + np+q , j1 ...jq

J = 1, . . . , n + np+q as local coordinates in a neighborhood π −1 (U ). If α ∈ qp (Mn ), is regarded, in a natural way, by contraction, as a function in Tqp (Mn ), which we denote by ıα. If α has local expression j ...j

α = αi11...ipq ∂j1 ⊗ . . . ⊗ ∂jq ⊗ dxi1 ⊗ . . . ⊗ dxip in a coordinate neighborhood U (xj ) ⊂ Mn , then ıα = α(t) has the local expression j ...j i ...i ıα = αi11...ipq tj11 ...jpq ¯

with respect to the coordinates (xj , xj ) in π −1 (U ). Suppose that A ∈ pq (Mn ). Then there is a unique vector field 1 0 (Tqp (Mn ))(vertical lift of A) such that for α ∈ qp (Mn ) [3] V

where

V

A(ıα) = α(A) ◦ π =V (α(A))

(α(A)) is the vertical lift of the function α(A) ∈ F (Mn ).

V

A∈ (5)

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¯

A =V Ak ∂k +V Ak ∂k¯ , then we have from (5) V

i ...i

¯

j ...j

k ...k

k ...k

h ...h

Ak tj11 ...jpq ∂k αi11...ipq +V Ak αh11 ...hqp = αh11 ...hqp Ak11...kqp

k ...k

j ...j

But αh11 ...hqp and ∂k αi11...ipq can take any preassigned values as each point. Thus, we have from the equation above V

¯

i ...i

h ...h

Ak tj11 ...jpq = 0,V Ak = Ak11...kqp .

Hence V

Ak = 0,

at all points of Tqp (Mn ) except possibly those at which all the components i ...i ¯ xj = tj11 ...jpq are zero: that is, at points of the base space. Thus we see that the ¯

components V Ak are zero a point such that xj = 0, that is, in Tqp (Mn ) → Mn . However, Tqp (Mn ) → Mn is dense in Tqp (Mn ) and the components of V A are continuous at every point of Tqp (Mn ). Hence, we have V Ak = 0 at all points of Tqp (Mn ). Consequently, the vertical lift V A of A to Tqp (Mn ) has components  V j   0 A V A= (6) = i ...i V ¯ Aj11 ...jpq Aj ¯

with respect to the coordinates (xj , xj ) in Tqp (Mn ). Let LV be the Lie derivation with respect to V ∈ 10 (Mn ) . We define ¯ V of V to Tqp (Mn ) [3] by the complete lift c V = L c

for α ∈ qp (Mn ).

V (ıα) = ı(LV α)

(7)

¯

If c V =c V k ∂k +c V k ∂k¯ , then we have from (7)

c

i ...i

¯

j ...j

k ...k

i ...i

j ...j

V k tj11 ...jpq ∂k αi11...ipq +c V k αh11 ...hqp = tj11 ...jpq (V k ∂k αi11...ipq −

q  μ=1

j ...k...jq

(∂k V jμ )αi11...ip

+

p  λ=1

(8) j ...j

q (∂iλ V k )αi11...k...i ) p

Thus, discussing in the same way as in the case of the vertical lift, from (8) we see that, c V has components c j   j V V c

q V = c ¯j = P (9) i1 ...m...ip i ...ip ∂m V iλ − μ=1 tj11 ...m...j ∂ Vm V λ=1 tj1 ...jq q jμ ¯

with respect to the coordinates (xj , xj ) in Tqp (Mn ). Suppose that there is given a tensor field ξ ∈ pq (Mn ). Then the correspondence x → ξx , ξx being the value of ξ at x ∈ Mn , determines a mapping σξ : Mn → Tqp (Mn ), such that π ◦ σξ = idMn , and the n-dimensional submanifold σξ (Mn ) of Tqp (Mn ) is called the cross-section determined by ξ. If the

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tensor field ξ has the local component ξk11...kqp (xk ), the cross-section σξ (Mn ) is locally expressed by  xk = xk (10) ¯ h ...h xk = ξk11...kqp (xk ) ¯

with respect to the coordinates (xk , xk ) in Tqp (Mn ). Differentiating (10) by xj , we see that n tangent vector fields Bj (j = 1, . . . , n) to σξ (Mn ) have components  δjk ∂xK K (Bj ) = ( j ) = (11) h ...h ∂j ξk11...kqp ∂x with respect to the natural frame {∂k , ∂k¯ } in Tqp (Mn ). On the other hand, the fibre is locally expressed by  xk = const, h1 ...hp h ...h tk1 ...kq = tk11...kqp , h ...h

tk11...kqp being considered as parameters. Thus, on differentiating with respect i ....i ¯ to xj = tj11 ...jqp , we see that np+q tangent vector fields C¯j (¯j = 1, . . . , np+q ) to the fibre have components  0 ∂xK K (12) (C¯j ) = ( ¯j ) = j h δkj11 . . . δkqq δih11 . . . δipp ∂x with respect to the natural frame {∂k , ∂k¯ } in Tqp (Mn ), where δij is the Kronecker symbol. Definition 2. A vector field X along a cross-section σξ : Mn → Tqp (Mn ) is mapping X : Mn → T (Tqp (Mn ))( T (Tqp (Mn ))- tangent bundle over the ∼ ∼ ∼ manifold Tqp (Mn )) such that π ◦x = σξ , where π is the projection π : T (Tqp (Mn )) → Tqp (Mn ). Thus X assigns to each point x ∈ Mn a tangent vector to Tqp (Mn ) at σξ (x) and therefore n + np+q local vector fields Bj and C¯j in π −1 (U ) ⊂ T p (Mn ) are vector fields along σξ (Mn ). They form a local family of frames  q Bj , C¯j along σξ (Mn ), which is called the adapted (B, C)- frame of σξ (Mn ) ¯ ¯ in π −1 (U ). From c V =c V h ∂h +c V h ∂h¯ and c V =c V j Bj +c V j C¯j , we easily ¯ ¯ ¯ ¯ ¯ obtain c V k =c V j Bjk +c V j C¯jk , c V k =c V j Bjk +c V j C¯jk . Now, taking account of (9) on the cross-section σξ (Mn ), and also (11) and (12), we have c V˜ k = V k , ¯ h ...h c ˜k V = −LV ξ 1 p . k1 ...kq

Thus, the complete lift c V has along σξ (Mn ) components of the form  c k   Vk V˜ c V = (13) = h ...h ¯ c ˜k −LV ξk11...kqp V

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with respect to the adapted (B, C)- frame. From (6), (11) and (12), by using similar way the vertical lift V A also has components  V k   0 A V A= (14) = h ...h ¯ V k Ak11...kqp A with respect to the adapted (B, C)- frame. Let ϕ ∈ 11 (Mn ). Making use of the Jacobian matrix of the coordinate transformation in Tqp (Mn ):  j  x = xj (xj ), ¯

i ...i

i

i

j

(i )

i ...i

(j)

¯

xj = tj1 ...jp = Ai11 . . . Aipp Ajj1 . . . Ajqq tj11 ...jpq = A(i) A(j  ) xj , q

1

1



∂xi ∂xj , Ajj1 = , we can i 1 1 ∂x ∂xj  1 p define a vector field γϕ ∈ 0 (Tq (Mn )), p ≥ 1, q ≥ 0:  0 J γϕ = ((γϕ) ) = li ...i tj12...jqp ϕil 1 (i )

i

i

(j)

j

i

where A(i) A(j  ) = Ai11 . . . Aipp Ajj1 . . . Ajqq , Ai11 =

where ϕil 1 are local components of ϕ in Mn . Clearly, we have γϕ(V f ) = 0 for any f ∈ F (Mn ). So that, γϕ is a vertical-vector lift of the tensor field ϕ ∈ 11 (Mn ) to Tqp (Mn ). We can easily verify that the vertical-vector lift γϕ has along σξ (Mn ) components  0 ∼ K γϕ = ((γϕ) ) = (15) lh ...h ξk12...kq p ϕhl 1 h ...h

with respect to the adapted (B, C)-frame, where ξk11...kqp are local components of ξ in Mn .

4. Interpretation of Complete Lifts of Affinors ∗

Let pq (Mn ) denote a module of all the tensor fields ξ ∈ pq (Mn ) which are pure with respect to ϕ. Now, we consider a pure cross-section σξϕ (Mn ) ∗

determined by ξ ∈ pq (Mn ), p ≥ 1, q ≥ 0. We observe that the local vector fields  h  ∂ δj c c c h ∂ X(j) = ( j ) = (δj )= 0 ∂x ∂xh V

¯

X (j) =V (∂j1 ⊗ . . . ⊗ ∂jp ⊗ dxi1 ⊗ . . . ⊗ dxiq ) i

k

=V (δhi11 . . . δhqq δjk11 . . . δjpp ∂k1 ⊗ . . . ⊗ ∂kp ⊗ dxh1 ⊗ . . . ⊗ dxhq )  0 = i k δhi11 . . . δhqq δjk11 . . . δjpp

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j = 1, . . . , n, ¯j = n + 1, . . . , n + np+q span the module of vector fields in π −1 (U ). Hence any tensor field is determined in π −1 (U ) by its action of ¯ c X(j) and V X (j) . Then we define a tensor field c ϕ ∈ 11 (Tqp (Mn )) along the pure cross-section σξϕ (Mn ) by  c c ϕ( V ) =c (ϕ(V )) − γ(LV ϕ) +V ((LV ϕ) ◦ ξ), ∀V ∈ 10 (Mn ), (i) (16) c ϕ(V A) =V (ϕ(A)), ∀A ∈ pq (Mn ), (ii) where ϕ(A) ∈ pq (Mn ), ((LV ϕ) ◦ ξ)(x1 , . . . , xq ; α1 , . . . , αp ) = ξ(x1 , . . . , xq ; (LV ϕ) α1 , . . . , αp ) and call c ϕ the complete lift of ϕ ∈ 11 (Mn ) to Tqp (Mn ), p ≥ 1, q ≥ 0 along σξϕ (Mn ). In particular, if we assume that p = 1, q > 0 then we get γ(LV ϕ) =V ((LV ϕ) ◦ ξ) substituting this into (16), we find (see [7]) c

ϕ(c V ) =c (ϕ(V )), c ϕ(V A) =V (ϕ(A))

Remark 4. The equation (16) is useful extension of the equation c L(ıα) = ı(LV α), α ∈ qp (Mn ) (see [3]) to affinor fields along the pure cross-section σξϕ (Mn ). c Let c ϕ˜K L be components of ϕ with respect to the adapted (B, C)- frame ϕ of the pure cross-section σξ (Mn ). Then, from (14), (15) and (16) we have  ∼ ∼ c K c ˜L ϕ˜L V =c (ϕ˜(V ))K − γ(LV ϕ)K +V ((LV ϕ) ◦ ξ)K , (i) (17) c K V ¨L ϕ˜L A =V (ϕ˜(A))K , (ii) where

 V

( (ϕ˜(A))K ) = V

mh ...hp



∼

((LV ϕ) ◦ ξ)K = ∼ γ(LV ϕ)K

0

 =

2 ϕhm1 Ak1 ...k q

,

0 h ...m...h (LV ϕhmλ )ξk11...kq p 0 h1 mh2 ...hp ((LV ϕ)m )ξk1 ...kq

, ,

LV ϕhmλ are local components of LV ϕ in Mn . First, consider the case where K = k. In the case, (i) of (17) reduces to c

¯ ϕ˜kl c V˜ l +c ϕ˜¯kl c V˜ l =c (ϕ˜(V ))k = (ϕ(V ))k = ϕkl V l .

(18)

Since the right-hand side of (18) are functions depending only on the base ¯ coordinates xi , the left-hand sides of (18) are too. Then, since c V l depend on fibre coordinates, from (18) we obtain c

ϕ˜¯kl = 0.

(19)

From (18) and (19), we have c ϕ˜kl c V˜ l =c ϕ˜kl V l = ϕkl V l , V i being arbitrary, which implies c ϕ˜kl = ϕkl .

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¯ (ii) of (17) reduces to When K = k, c

˜

mh ...hp

r

...sp = δkr11 . . . δkqq ϕhs11 δsh22 . . . δshpp Asr11 rs22 ...r q

2 ...sp ϕ˜˜kl Asr11 ...r = ϕhm1 Ak1 ...k q q

for all A ∈ pq (Mn ), which implies c ¯

¯

s ...s

¯

r

ϕ˜¯kl = δkr11 . . . δkqq ϕhs11 δsh22 . . . δshpp h ...h

where xl = tr11 ...rqp , xk = tk11...kqp . ¯ (i) of (17) reduces to When K = k, c

¯ ¯ ¯ ¯ ϕ˜kl c V˜ l +c ϕ˜¯kl c V˜ l =c (ϕ˜(V ))k −

p  λ=2

h ...l...hp

(LV ϕhl λ )ξk11...kq

or c

¯ ϕ˜kl c V˜ l

+

ϕhs11 δsh22

. . . δshpp δkr11

r c ¯ . . . δkqq V˜ l +

p  λ=2

h h ...l...hp

2 (LV ϕhl λ )ξk11...k q

¯ =c (ϕ˜(V ))k .

(20) ¯

Now, using the Φϕ –operator we will investigate components c ϕ˜kl . The ∗

Φϕ -operator on the pure module pq (Mn ) is given by h ...h

∗

h ...h

h ...h

1 p 1 p (Φϕ ξ)lk11 ...kpq = ϕm l ∂m ξk1 ...kq − ∂l ξk1 ...kq

+

q  a=1

h ...h

1 p (∂ka ϕm l ) ξk1 ...m...kq + 2

p  λ=1

h ...m...hp

λ ∂[l ϕhm] ξk11...kq

,

where h ...h

1 p ϕm k1 ξmk2 ...kq

h m...hp

ϕhm2 ξk11...kq

h ...h

h ...h

mh ...hp

1 p 1 p 2 m h1 = ϕm k2 ξk1 m...kq = . . . = ϕkq ξk1 k2 ...m = ϕm ξk1 ...kq

h2 ...m = . . . = ϕhmp ξkh11...k = q

=

∗ h h ...h ξk11k22...kqp .

After some calculations we have h ...h

∗

h ...h

h ...h

V l (Φϕ ξ)lk11 ...kpq = (LϕV ξ)k11...kqp − (LV ξ)k11...kqp +

p  λ=1

h ...m...hp

(LV ϕhmλ )ξk11...kq

or h ...h

mh ...hp

2 V l (Φϕ ξ)lk11 ...kpq + ϕhm1 (LV ξ)k1 ...k q

−

p  λ=1

for any V ∈ 10 (Mn ).

mh ...h

2 p + ξk1 ...k (LV ϕ)hm1 q

h ...m...hp

(LV ϕhmλ )ξk11...kq

h ...h

= (LϕV ξ)k11...kqp

(21)

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Using (13), from (21) we have h ...h

mh ...hp

2 V l (Φϕ ξ)lk11 ...kpq + ϕhm1 (LV ξ)k1 ...k q

−

p  λ=1

mh ...h

2 p + ξk1 ...k (LV ϕhm1 ) q

h ...m...hp

(LV ϕhmλ )ξk11...kq h ...h

r

...sp = V l (Φϕ ξ)lk11 ...kpq + ϕhs11 δsh22 . . . δshpp δkr11 . . . δkqq (LV ξ)sr11 ...r q

−

p  λ=2

h h ...m...hp

2 (LV ϕhmλ )k11...k q

h ...h

r c

¯

= V l (Φϕ ξ)lk11 ...kpq − ϕhs11 δsh22 . . . δshpp δkr11 . . . δkqq V l c

−

p  λ=2

h h ...m...hp

2 (LV ϕhmλ )ξk11...k q

¯

= −c (ϕ(V ))k

or h ...h

r

¯

− (Φϕ ξ)lk11 ...kpq c V l + ϕhs11 δsh22 . . . δshpp δkr11 . . . δkqq c V l +

p  λ=2

h h ...m...hp

2 (LV ϕhmλ )ξk11...k q

¯

=c (ϕ(V ))k .

(22)

Comparing (20) and (22), we get c

¯

h ...h

ϕ˜kl = −(Φϕ ξ)lk11 ...kpq .

Thus we have Theorem 2. Let ϕ ∈ 11 (Mn ) and σξϕ be a pure cross-section of Tqp (Mn ) with respect to ϕ. Then the complete lift c ϕ ∈ 11 (Tqp (Mn )) of ϕ has along the pure cross-section σξϕ (Mn ) components  ¯ h ...h c k ϕ˜l = ϕkl , c ϕ˜¯kl = 0, c ϕ˜kl = −(Φϕ ξ)lk11 ...kpq , (23) ¯ h r c k ϕ˜¯l = ϕhs11 δsh22 . . . δspp δkr11 . . . δkqq with respect to the adapted (B, C)- frame of σξϕ (Mn ), where Φϕ ξ is the Φϕ ¯

h ...h

¯

s ...s

operator and xk = tk11...kqp , xl = tr11 ...rqp .

∗

Remark 5. c ϕ in the form (23) is unique solution of (16). Therefore, if ϕ is ∗ element of 11 (Tqp (Mn )), such that ϕ(c V ) =c ϕ(c V ) =c (ϕ(V )) − γ(LV ϕ) +V ∗

∗

((LV ϕ) ◦ ξ), ϕ(V A) =c ϕ(V A) =V (ϕ(A)), then ϕ =c ϕ.

5. Almost Hyperholomorphic Pure Submanifolds in the Tensor Bundle On putting B¯j = C¯j , we write the adapted (B, C)- frame of σξϕ (Mn ) as   ˜ J of σ ϕ (Mn ) by B ˜ I (BJ ) = δ I . From BJ = Bj , B¯j . We define a coframe B J ξ

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˜ I = δ I we see that covector fields B ˜ I have components BJK B K J ˜ i ) = (δ i , 0) ˜ i = (B B K k j ...j k j ¯ ı ˜ ¯ı = (B ˜K B ) = (−∂k ξi11...iqp , δik11 . . . δiqq δhj11 . . . δhpp )

(24)

¯

with respect to the natural coframe (dxk , dxk ). Taking account of c

ϕK L =

c

=

˜ I (dxk , ∂L ) ϕ(dxk , ∂L ) =c ϕ˜JI BJ ⊗ B c J ˜ I (∂L ) =c ϕ˜J dxK (B H ∂H )B ˜I ϕ˜ dxK (BJ )B

=

c

I

I

K ˜I BL ϕ˜JI BJH δH

=

c

J

L

˜LI , ϕ˜JI BJK B

and also (23) and (24), we see that ϕ has along the pure cross-section σξϕ (Mn ) components of the form c

c

ϕkl = ϕkl , c ϕ¯kl = 0,

c

ϕ¯kl = ϕhs11 δsh22 . . . δshpp δkr11 . . . δkqq ,

c

2 ϕkl = (∂l ϕhm1 )ξk1 ...k q

¯

r

¯

mh ...hp

−

p  λ=1

−

q  μ=1

h ...h

1 p (∂kμ ϕm l )ξk1 ...m...kq

(25)

h ...m...hp

(∂l ϕhmλ − ∂m ϕhl λ )ξk11...kq

with respect to the natural frame {∂h , ∂h¯ } of σξϕ (Mn ) in π −1 (U ) [4].   By using (25), in [4] proved that if Π = ϕ is almost integrable algeα   braic hypercomplex Π-structure on Mn , then the complete lift c Π = c ϕ α

of Π to Tqp (Mn ) along the pure cross-section σξΠ (Mn ) is an algebraic hypercomplex c Π-structure on Tqp (Mn ). For an element X ∈ 10 (Mn ) with local coordinates X h , we denote by BX and CX the vector fields with local components  Xk = X j Bj , BX = h ...h X j ∂j ξk11...kqp  0 CX = k s δrk11 . . . δrqq X h1 δhs22 . . . δhpp  0 s = δrk11 . . . δrkqq X i1 δhs22 . . . δhpp j h δkj11 . . . δkqq δih11 . . . δipp s

= δrk11 . . . δrkqq X i1 δhs22 . . . δhpp C(¯j) which are tangent to σξ (Mn ) and the fibre, respectively. Then by (23), we ∗

have along the pure cross-section σξ (Mn ) determined by ξ ∈ pq (Mn ) that c

ϕ(BX) = B(ϕX) − C((Φϕ ξ)(X; Y1 , . . . , Yq , ξ 1 , . . . , ξ p ))

(26)

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for any X ∈ 10 (Mn ). When c ϕ(BX) is tangent to σξ (Mn ) for any X ∈ 10 (Mn ), c ϕ is said to leave σξ (Mn ) invariant. We have from (26) Theorem 3. The complete lift c ϕ of an element ϕ ∈ 11 (Mn ) leaves the pure cross-section σξ (Mn ) invariant if and only if Φϕ ξ = 0. A submanifold in an almost algebraic hypercomplex manifold with struc  ture Π =

ϕ , is said to be almost hyperholomorphic when ϕ, α = 1, . . . , m α

α

leaves the submanifold invariant. Thus, from Remark 3 and Theorem 3, we have

Theorem 4. A necessary and sufficient condition for the pure cross-section σξ (Mn ) in Tqp (Mn ) determined by a pure tensor field ξ ∈ pq (Mn ) in an   almost hypercomplex manifold Mn with structure Π = ϕ to be almost α

hyperholomorphic submanifoldin  the almost algebraic hypercomplex manifold Tqp (Mn ) with structure c Π = c ϕ is that the pure tensor field ξ ∈ pq (Mn ) α

be almost hyperholomorphic in Mn .

References [1] L. E. Evtushik, Yu. G. Lumiste, N. M. Ostianu, A. P. Shirokov, Differentialgeometric structures on manifolds. J. Sov. Math. 14 (1980), 1573-1719. [2] G. I. Kruckovic, Hyperkomplexe Strukturen auf Mannigfaltigkeiten. I. (Russian) Tr. Sem. vektor. tenzor. Analizu 16 (1972), 174-201. [3] A. J. Ledger, K. Yano, Almost complex structures on tensor bundles. J. Differ. Geom. 1 (1967), 355-368. [4] A. A. Salimov, Quasiholomorphic mapping and tensor fibering. Sov. Math. 33 (12) (1989), 89-92. [5] A. A. Salimov, Almost ψ-holomorphic tensors and their properties. Russ. Acad. Sci., Dokl., Math. 45 No.3 (1992), 602-605. [6] A. A. Salimov, The lifts of polyaffinor structures on the pure sections of a tensor bundle. Russ. Math. 40 No. 10 (1996), 52-59. [7] A. A. Salimov, A. Ma˘ gden, Complete lifts of tensor fields on a pure crosssection in the tensor bundle. Note Mat. 18 No.1 (1998), 27-37. [8] A. A. Salimov, M. Iscan, F. Etayo, Paraholomorphic B-manifold and its properties. Topology Appl. 154 No. 4 (2007), 925-933. [9] A. A. Salimov, M. Iscan, K. Akbulut, Some remarks concerning hyperholomorphic B-manifolds. Chin. Ann. Math., Ser. B 29 No. 6 (2008), 631-640. [10] G. Scheffers, Verallgemeinerung der Grundlagen der gew¨ o hnlich complexen Functionen. I. Leipz. Ber. XLV. (1893), 828-848. [11] V. V. Vishnevskii, A. P. Shirokov, V. V. Shurygin, Spaces over algebras. (Prostranstva nad algebrami). (Russian)Kazan’: Izdatel’stvo Kazanskogo Universiteta. (1985), p. 263.

Vol. 22 (2012)

Applications of Φ-Operators to Hypercomplex Geometry

Arif Salimov1,2 and Seher Aslanci2 1 Baku State University Department of Algebra and Geometry AZ-1148 Baku Azerbaijan 2

Ataturk University Faculty of Sciences Department of Mathematics 25240 Erzurum Turkey e-mail: [email protected] [email protected] Received: October 15, 2010. Accepted: December 15, 2010.

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