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Horizontal lift of affinor structures and its applications
Applied Mathematics and Computation 156 (2004) 455–461 www.elsevier.com/locate/amc
Horizontal lift of affinor structures and its applications A. Ma gden, N. Cengiz *, A.A. Salimov Department of Mathematics, Faculty of Art and Sciences, Atat€urk University, 25240 Erzurum, Turkey
Abstract The main purpose of the present paper is to study the horizontal lifts of tensor field of type (1, 1) (affinor field) to tensor bundle and the integrability conditions for the horizontal lifts of special types of complex and tangent structures. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Horizontal lift; Nijenhuis tensor; K€ahlerian manifol; B-manifold
1. Introduction Let Mn be a differentiable manifold of class C 1 and finite dimension n, and let Tqp ðMnÞ, p þSq P 1 be the bundle over Mn of tensors of type ðp; qÞ : Tqp ðMn Þ ¼ P 2Mn Tqp ðP Þ, where Tqp ðP Þ denotes the tensor spaces of tensors of type ðp; qÞ at P 2 Mn . We list below notations used is this paper: ii(i) p : Tqp ðMn Þ ! Mn is the natural projection Tqp ðMn Þ onto Mn . i(ii) The indices i; j; . . . run from 1 to n, the indices i; j; . . . ; from n þ 1 to n þ npþq ¼ dim Tqp ðMn Þ and the indices I ¼ ði; iÞ, J ¼ ðj; jÞ; . . . from 1 to n þ npþq . (iii) F ðMn Þ is the ring of real-valued C 1 functions on Mn . Tpq ðMn Þ is the module over F ðMn Þ of C 1 tensor fields of type ðp; qÞ.
*
Corresponding author. E-mail address: [email protected] (N. Cengiz).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.004
456
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
Denoting by xj the local coordinates of p ¼ pð~pÞ ð~p 2 Tqp ðMn ÞÞ in a neighi ...i borhood U Mn and if we make ðxj ; tj11 ...jpp Þ ¼ ðxj ; xj Þ correspond to the point 1 ~ p 2 p ðU Þ, we can introduce a system of local coordinates ðxj ; xj Þ in a i ...i neighborhood p 1 ðU Þ Tqp ðMn Þ, where tj11 ...jpq ¼ xj are components p of t 2 Tq ðMn Þ with respect to the natural frame. If a 2 Tqp ðMn Þ, it is regarded, in a natural way (by contraction), as a function in Tqp ðMnÞ, which we denote by ia. If a has the local expression j ...j a ¼ ai11...ipq @j1 . . . @jq dxi1 . . . dxip in a coordinate neighborhood U Mn , then ia has the local expression j ...j i ...i
ia ¼ aðtÞ ¼ ai11...ipq tj11 ...jpq
with respect to the coordinates ðxj ; xj Þ in p 1 ðU Þ. Let A 2 Tpq ðMn Þ. We define the vertical lift V A 2 T10 ðTqp ðMn ÞÞ of A to Tqp ðMn Þ by [4] V
AðiaÞ ¼ aðAÞ p ¼ V ðaðAÞÞ;
where V ðaðAÞÞ is the vertical lift of the function aðAÞ 2 F ðMn Þ. The vertical lift A of A to Tqp ðMn Þ has components V j 0 A V i1 ...ip A ¼ V j ¼ Aj1 ...jq A
V
with respect to the coordinates ðxj ; xj Þ in Tqp ðMn Þ. Suppose now that r is a linear connection (with zero torsion) on Mn , and let X 2 T10 ðMn Þ. We define the horizontal lift H X 2 T10 ðTqp ðMn ÞÞ of X to Tqp ðMn Þ by [4] H
X ðiaÞ ¼ iðrX aÞ;
where a 2 Tqp ; ðMn Þ, rX is the covariant differentiation with respect to the vector field X . The horizontal lift H X of X to Tqp ðMn Þ has components ! q p X X l1 ...lp s lk l1 ...s...lp H k k H k m X ¼X ; X ¼X Cmkl tk1 ...s...kq Cms tk1 ...kq ; l¼1
k¼1
Chij
are components of r in Mn . where Let A, B 2 Tpq ðMn Þ and X , Y 2 T10 ðMn Þ and let F 2 T11 ðMn Þ. Let R denote the curvature tensor field of the connection r. Then (see [1,4]) 8 V V ½ A; B ¼ 0; > > < H V ½ X ; A ¼ V ðrX AÞ; ð1Þ > ½H X ; ð~c cÞF ¼ ð~c cÞðrX F Þ; > : H H ½ X ; Y ¼ H ½X ; Y þ ð~c cÞRðX ; Y Þ; where ð~c cÞF is a vector field in Tqp ðMn Þ defined by
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
ð~c cÞF ¼
q X
i ...i
p tj11 ...m...j Fm q jl
l¼1
p X
! i ...m...i
tj11 ...jq p Fmik
k¼1
457
o oxj
with respect to the coordinates ðxj ; xj Þ.
2. Horizontal lifts of tensor fields of type (1, 1) We shall first prove the following lemma, which is useful in determining tensor fields in tensor bundle. Lemma. Let e S and Te be tensor fields in Tqp ðMn Þ of type ð1; qÞ, where q > 0, such that e eq Þ ¼ Te ð X e1 ; . . . ; X eq Þ e1 ; . . . ; X S ðX eq which are of the form V A or H Z, where A 2 Tp ðMn Þ e1 ; . . . ; X for all vector fields X q 1 and Z 2 T0 ðMn Þ. Then e S ¼ Te . Proof. It is sufficient to prove that there are n þ npþq linearly independent vector fields of the form H Zj and V Aj ; j ¼ 1; . . . ; n; j ¼ n þ 1; . . . ; n þ npþq in each coordinate neighborhood p 1 ðU Þ of Tqp ðMn Þ, where Zj 2 T10 ðMn Þ and Aj 2 Tpq ðMn Þ. The vector fields H Zj ¼ H ðo=oxj Þ and V Aj ¼ V ðol1 . . . olp
dxj1 . . . dxjq Þ, which have, respectively, components of the form ! dhj H Pq Zj ¼ Pq k1 ...kp kk k1 ...s...kp s l¼1 Cjhl th1 ...s...hq k¼1 Cjs th1 ...hq and V
Aj ¼
0
dkl11
k . . . dlpp djh11
j . . . dhqq
k ...k
with respect to the coordinates ðxh ; xh Þ, xh ¼ th11 ...hpq in p 1 ðU Þ Tqp ðMn Þ, where dij Kronecker delta, are n þ npþq linearly independent vector fields in p 1 ðU Þ Tqp ðMn Þ and span the module of vector fields in p 1 ðU Þ Tqp ðMn Þ. Thus Lemma is proved. h Let u 2 T11 ðMn Þ. We put
H H uð X Þ ¼ H ðuðX ÞÞ; 8X 2 T10 ðMn Þ; H uðV AÞ ¼ V ðuðAÞÞ; 8A 2 Tpq ðMn Þ;
ð2Þ
458
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
where uðAÞ ¼ Cðu AÞ 2 Tpq ðMn Þ. We call H u the horizontal lift of u to Tqp ðMn Þ. According to Lemma, the horizontal lift H u of u is completely determined by (2). The horizontal lift H u has components of the form ( 8 r uls11 dls22 . . . dlspp drk11 . . . dkqq ; p P 1; > k > H k k H k H > u ¼ u ; u ¼ 0; u ¼ > l r l l l > dls11 . . . dlspp urk11 drk22 . . . dkqq ; q P 1 > > > P > Pq > l1 ...s...lp ðlÞ p H k > > ul ¼ uml k¼1 Cllsk nðkÞ þ l¼1 Csmkl nk1 ...s...kq > > P > Pp < ml2 ...lp ml ...s...lp s...l q þulm1 l¼1 Cslkl nk1 ...s...k þ k¼2 Cllsk nðkÞ2 þ Cmls nðkÞ p ; q > > p P1 > > P P > l1 ...s...lp ðlÞ > H k > u ¼ uml pk¼1 Clmsk nðkÞ þ ql¼1 Csmkl nk1 ...s...kq > l > P > > Pq ðlÞ ðlÞ p > lk l1 ...s...lp s s m > þu C n C n C n > k¼1 ls mk2 ...kq l¼2 lkl mk2 ...s...kq lm m...kq ; k1 > : qP1 with respect to the coordinates ðxj ; xj Þ in p 1 ðU Þ Tqp ðMn Þ (see [2]). Theorem 1. Let u be an almost complex (almost tangent) structure in Mn with symmetric affine connection r. Then H u is an almost complex (almost tangent) structure in Tqp ðMn Þ. Proof. If A 2 Tpq ðMn Þ, then by (2). We find 2
V
ðH uÞ V A ¼ H uðH uV AÞ ¼ H uV ðuðAÞÞ ¼ V ðu2 AÞ ¼ H ðu2 Þ A;
ð3Þ
If X 2 T10 ðMn Þ, then, by (2), we find ðH uÞ2 H X ¼ H uðH uH X Þ ¼ H uH ðuðX ÞÞ ¼ H ðu2 X Þ ¼ H ðu2 ÞH X ;
ð4Þ
Since H ðidMn Þ ¼ idTqp ðMn Þ , then the required result follows from (3) and (4) because of Lemma. h Theorem 2. If F , G 2 T11 ðMn Þ, then H
F ð~c cÞG ¼ ð~c cÞðGF Þ:
ð5Þ
Proof. (5) will be established by using local expressions of H F and cG. Let F 2 T11 ðMn Þ and NF be the Nijenhuis tensor of F : NF ðX ; Y Þ ¼ ½FX ; FY F ½FX ; Y F ½X ; FY þ F 2 ½X ; Y ;
X ; Y 2 T10 ðMn Þ:
e H F be a the Nijenhuis tensor of H F in T p ðMn Þ. Then, by (1) and (5), if Let now N q 1 X ; Y 2 T0 ðMn Þ, A; B; 2 Tpq ðMn Þ, we have
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
459
e H F ðV A; V BÞ N 2
V
¼ ½H F V A; H F V B H F ½H F V A; V B H F ½ A; H F V B þ ðH F Þ ½V A; V B 2
¼ ½V ðFAÞ; V ðFBÞ H F ½V ðFAÞ; V B H F ½V A; V ðFBÞ þ ðH F Þ ½V A; V B ¼ 0; e H F ðH X ; V BÞ N 2
¼ ½H F H X ; H F V B H F ½H F H X ; V B H F ½H X ; H F V B þ ðH F Þ ½H X ; V B 2
¼ ½H ðFX Þ; V ðFBÞ H F ½H ðFX Þ; V B H F ½V X ; V ðFBÞ þ ðH F Þ ½H X ; V B 2
¼ V ðrFX F ðBÞÞ H F V ðrFX BÞ H F V ðrX F ðBÞÞ þ ðH F Þ V ðVX BÞ ¼ V ðrFX F ðBÞÞ V ðF ðrFX BÞÞ V ðF ðrX F ðBÞÞÞ þ V ðF 2 ðrX BÞÞ ¼ V frFX F ðBÞ F ðrFX BÞ F ðrX F ðBÞÞ þ F 2 ðrX BÞg ¼ V fðrFX ÞB ðF ðrX F ÞÞBg; e H F ðH X ; H Y Þ N 2
¼ ½H F H X ; H F H Y H F ½H F H X ; H Y H F ½H X ; H F H Y þ ðH F Þ ½H X ; H Y ¼ H ½FX ; FY ð~c cÞRðFX ; FY Þ H F H ½ðFX Þ; Y H F ð~c cÞRðFX ; Y Þ 2
H F H ½X ; FY H F ð~c cÞRðX ; FY Þ þ ðH F Þ H ½X ; Y þ ðH F Þð~c cÞRðX ; Y Þ ¼ H ðNF ðX ; Y ÞÞ þ ð~c cÞfRðFX ; FY Þ RðFX ; Y Þ RðX ; FY Þ þ RðX ; Y ÞF 2 g: Summing up, we have the following formulas: 8 e H F ðV A; V BÞ ¼ 0; > N > > > N > > > : RðX ; FY Þ þ RðX ; Y ÞF 2 g:
ð6Þ
3. Applications 1. We now suppose that ðF ; gÞ is a K€ ahlerian structure in Mn and r the Riemannian connection determined by the metric g. Then we see that ii(i) F is an almost complex structure in Mn , i.e., F 2 ¼ I; i(ii) rF ¼ 0; (iii) The curvature tensor R of r satisfies RðFX ; FY Þ ¼ RðX ; Y Þ;
8X ; Y 2 T10 ðMn Þ:
460
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
From (iii) we get RðFX ; Y Þ ¼ RðX ; FY Þ; i.e. RðX ; Y Þ hybrid in X and Y . Hence using F 2 ¼ I, we find RðFX ; FY Þ RFX ; Y ÞF RðX ; FY ÞF þ RðX ; Y ÞF 2 ¼ 0: Therefore it follows, from (6) and (ii), that 8 e H F ðV A; V BÞ ¼ 0;
ð7Þ
i.e. curvature tensor R is pure in all arguments [3]. From (i) and (7) we get RðFX ; FY Þ ¼ RðX ; F 2 Y Þ ¼ 0;
ð8Þ
ðRðFX ; Y Þ þ RðX ; FY ÞÞF ¼ 2RðFX ; Y ÞF ¼ 2FRðFX ; Y Þ ¼ 2RðF 2 X ; Y Þ ¼ 0: ð9Þ Thus from (6), (8) and (iii) we have e H F ðV A; V BÞ ¼ 0; N
e H F ðH X ; V BÞ ¼ 0; N
e H F ðH X ; H Y Þ ¼ 0: N
e is zero, since N is skew-symmetric. Thus H F is necessarily Hence, by Lemma N integrable. Summing up, we have
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
461
Theorem 4. Let ðMn ; gÞ be a B-manifold with tangent structure F 2 ¼ 0 and r the Riemannian connection determined by g. Then the horizontal lift H F of F to Tqp ðMn Þ with respect to r is a tangent structure.
References [1] N. Cengiz, A.A. Salimov, Complete lifts of derivations to tensor bundles, Bol. Soc. Mat. Mexicana 8 (3) (2002) 75–82. [2] A. Ma gden, A.A. Salimov, Horizontal lifts of tensor fields to sections of a tensor bundle, Izv. Vyssh. Uchebu. Zaved. Mat. 45 (3) (2001) 73–76. [3] V.V. Vishnevskii, Affinor structures of affinely connected, Izv. Vyssh. Uchebu. Zaved. Mat 1 (1970) 12–23. [4] A. Ledger, K. Yano, Almost complex structures on tensor bundles, J. Dif. Geom. 1 (1967) 355– 368.
Horizontal lift of affinor structures and its applications A. Ma gden, N. Cengiz *, A.A. Salimov Department of Mathematics, Faculty of Art and Sciences, Atat€urk University, 25240 Erzurum, Turkey
Abstract The main purpose of the present paper is to study the horizontal lifts of tensor field of type (1, 1) (affinor field) to tensor bundle and the integrability conditions for the horizontal lifts of special types of complex and tangent structures. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Horizontal lift; Nijenhuis tensor; K€ahlerian manifol; B-manifold
1. Introduction Let Mn be a differentiable manifold of class C 1 and finite dimension n, and let Tqp ðMnÞ, p þSq P 1 be the bundle over Mn of tensors of type ðp; qÞ : Tqp ðMn Þ ¼ P 2Mn Tqp ðP Þ, where Tqp ðP Þ denotes the tensor spaces of tensors of type ðp; qÞ at P 2 Mn . We list below notations used is this paper: ii(i) p : Tqp ðMn Þ ! Mn is the natural projection Tqp ðMn Þ onto Mn . i(ii) The indices i; j; . . . run from 1 to n, the indices i; j; . . . ; from n þ 1 to n þ npþq ¼ dim Tqp ðMn Þ and the indices I ¼ ði; iÞ, J ¼ ðj; jÞ; . . . from 1 to n þ npþq . (iii) F ðMn Þ is the ring of real-valued C 1 functions on Mn . Tpq ðMn Þ is the module over F ðMn Þ of C 1 tensor fields of type ðp; qÞ.
*
Corresponding author. E-mail address: [email protected] (N. Cengiz).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.004
456
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
Denoting by xj the local coordinates of p ¼ pð~pÞ ð~p 2 Tqp ðMn ÞÞ in a neighi ...i borhood U Mn and if we make ðxj ; tj11 ...jpp Þ ¼ ðxj ; xj Þ correspond to the point 1 ~ p 2 p ðU Þ, we can introduce a system of local coordinates ðxj ; xj Þ in a i ...i neighborhood p 1 ðU Þ Tqp ðMn Þ, where tj11 ...jpq ¼ xj are components p of t 2 Tq ðMn Þ with respect to the natural frame. If a 2 Tqp ðMn Þ, it is regarded, in a natural way (by contraction), as a function in Tqp ðMnÞ, which we denote by ia. If a has the local expression j ...j a ¼ ai11...ipq @j1 . . . @jq dxi1 . . . dxip in a coordinate neighborhood U Mn , then ia has the local expression j ...j i ...i
ia ¼ aðtÞ ¼ ai11...ipq tj11 ...jpq
with respect to the coordinates ðxj ; xj Þ in p 1 ðU Þ. Let A 2 Tpq ðMn Þ. We define the vertical lift V A 2 T10 ðTqp ðMn ÞÞ of A to Tqp ðMn Þ by [4] V
AðiaÞ ¼ aðAÞ p ¼ V ðaðAÞÞ;
where V ðaðAÞÞ is the vertical lift of the function aðAÞ 2 F ðMn Þ. The vertical lift A of A to Tqp ðMn Þ has components V j 0 A V i1 ...ip A ¼ V j ¼ Aj1 ...jq A
V
with respect to the coordinates ðxj ; xj Þ in Tqp ðMn Þ. Suppose now that r is a linear connection (with zero torsion) on Mn , and let X 2 T10 ðMn Þ. We define the horizontal lift H X 2 T10 ðTqp ðMn ÞÞ of X to Tqp ðMn Þ by [4] H
X ðiaÞ ¼ iðrX aÞ;
where a 2 Tqp ; ðMn Þ, rX is the covariant differentiation with respect to the vector field X . The horizontal lift H X of X to Tqp ðMn Þ has components ! q p X X l1 ...lp s lk l1 ...s...lp H k k H k m X ¼X ; X ¼X Cmkl tk1 ...s...kq Cms tk1 ...kq ; l¼1
k¼1
Chij
are components of r in Mn . where Let A, B 2 Tpq ðMn Þ and X , Y 2 T10 ðMn Þ and let F 2 T11 ðMn Þ. Let R denote the curvature tensor field of the connection r. Then (see [1,4]) 8 V V ½ A; B ¼ 0; > > < H V ½ X ; A ¼ V ðrX AÞ; ð1Þ > ½H X ; ð~c cÞF ¼ ð~c cÞðrX F Þ; > : H H ½ X ; Y ¼ H ½X ; Y þ ð~c cÞRðX ; Y Þ; where ð~c cÞF is a vector field in Tqp ðMn Þ defined by
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
ð~c cÞF ¼
q X
i ...i
p tj11 ...m...j Fm q jl
l¼1
p X
! i ...m...i
tj11 ...jq p Fmik
k¼1
457
o oxj
with respect to the coordinates ðxj ; xj Þ.
2. Horizontal lifts of tensor fields of type (1, 1) We shall first prove the following lemma, which is useful in determining tensor fields in tensor bundle. Lemma. Let e S and Te be tensor fields in Tqp ðMn Þ of type ð1; qÞ, where q > 0, such that e eq Þ ¼ Te ð X e1 ; . . . ; X eq Þ e1 ; . . . ; X S ðX eq which are of the form V A or H Z, where A 2 Tp ðMn Þ e1 ; . . . ; X for all vector fields X q 1 and Z 2 T0 ðMn Þ. Then e S ¼ Te . Proof. It is sufficient to prove that there are n þ npþq linearly independent vector fields of the form H Zj and V Aj ; j ¼ 1; . . . ; n; j ¼ n þ 1; . . . ; n þ npþq in each coordinate neighborhood p 1 ðU Þ of Tqp ðMn Þ, where Zj 2 T10 ðMn Þ and Aj 2 Tpq ðMn Þ. The vector fields H Zj ¼ H ðo=oxj Þ and V Aj ¼ V ðol1 . . . olp
dxj1 . . . dxjq Þ, which have, respectively, components of the form ! dhj H Pq Zj ¼ Pq k1 ...kp kk k1 ...s...kp s l¼1 Cjhl th1 ...s...hq k¼1 Cjs th1 ...hq and V
Aj ¼
0
dkl11
k . . . dlpp djh11
j . . . dhqq
k ...k
with respect to the coordinates ðxh ; xh Þ, xh ¼ th11 ...hpq in p 1 ðU Þ Tqp ðMn Þ, where dij Kronecker delta, are n þ npþq linearly independent vector fields in p 1 ðU Þ Tqp ðMn Þ and span the module of vector fields in p 1 ðU Þ Tqp ðMn Þ. Thus Lemma is proved. h Let u 2 T11 ðMn Þ. We put
H H uð X Þ ¼ H ðuðX ÞÞ; 8X 2 T10 ðMn Þ; H uðV AÞ ¼ V ðuðAÞÞ; 8A 2 Tpq ðMn Þ;
ð2Þ
458
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
where uðAÞ ¼ Cðu AÞ 2 Tpq ðMn Þ. We call H u the horizontal lift of u to Tqp ðMn Þ. According to Lemma, the horizontal lift H u of u is completely determined by (2). The horizontal lift H u has components of the form ( 8 r uls11 dls22 . . . dlspp drk11 . . . dkqq ; p P 1; > k > H k k H k H > u ¼ u ; u ¼ 0; u ¼ > l r l l l > dls11 . . . dlspp urk11 drk22 . . . dkqq ; q P 1 > > > P > Pq > l1 ...s...lp ðlÞ p H k > > ul ¼ uml k¼1 Cllsk nðkÞ þ l¼1 Csmkl nk1 ...s...kq > > P > Pp < ml2 ...lp ml ...s...lp s...l q þulm1 l¼1 Cslkl nk1 ...s...k þ k¼2 Cllsk nðkÞ2 þ Cmls nðkÞ p ; q > > p P1 > > P P > l1 ...s...lp ðlÞ > H k > u ¼ uml pk¼1 Clmsk nðkÞ þ ql¼1 Csmkl nk1 ...s...kq > l > P > > Pq ðlÞ ðlÞ p > lk l1 ...s...lp s s m > þu C n C n C n > k¼1 ls mk2 ...kq l¼2 lkl mk2 ...s...kq lm m...kq ; k1 > : qP1 with respect to the coordinates ðxj ; xj Þ in p 1 ðU Þ Tqp ðMn Þ (see [2]). Theorem 1. Let u be an almost complex (almost tangent) structure in Mn with symmetric affine connection r. Then H u is an almost complex (almost tangent) structure in Tqp ðMn Þ. Proof. If A 2 Tpq ðMn Þ, then by (2). We find 2
V
ðH uÞ V A ¼ H uðH uV AÞ ¼ H uV ðuðAÞÞ ¼ V ðu2 AÞ ¼ H ðu2 Þ A;
ð3Þ
If X 2 T10 ðMn Þ, then, by (2), we find ðH uÞ2 H X ¼ H uðH uH X Þ ¼ H uH ðuðX ÞÞ ¼ H ðu2 X Þ ¼ H ðu2 ÞH X ;
ð4Þ
Since H ðidMn Þ ¼ idTqp ðMn Þ , then the required result follows from (3) and (4) because of Lemma. h Theorem 2. If F , G 2 T11 ðMn Þ, then H
F ð~c cÞG ¼ ð~c cÞðGF Þ:
ð5Þ
Proof. (5) will be established by using local expressions of H F and cG. Let F 2 T11 ðMn Þ and NF be the Nijenhuis tensor of F : NF ðX ; Y Þ ¼ ½FX ; FY F ½FX ; Y F ½X ; FY þ F 2 ½X ; Y ;
X ; Y 2 T10 ðMn Þ:
e H F be a the Nijenhuis tensor of H F in T p ðMn Þ. Then, by (1) and (5), if Let now N q 1 X ; Y 2 T0 ðMn Þ, A; B; 2 Tpq ðMn Þ, we have
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
459
e H F ðV A; V BÞ N 2
V
¼ ½H F V A; H F V B H F ½H F V A; V B H F ½ A; H F V B þ ðH F Þ ½V A; V B 2
¼ ½V ðFAÞ; V ðFBÞ H F ½V ðFAÞ; V B H F ½V A; V ðFBÞ þ ðH F Þ ½V A; V B ¼ 0; e H F ðH X ; V BÞ N 2
¼ ½H F H X ; H F V B H F ½H F H X ; V B H F ½H X ; H F V B þ ðH F Þ ½H X ; V B 2
¼ ½H ðFX Þ; V ðFBÞ H F ½H ðFX Þ; V B H F ½V X ; V ðFBÞ þ ðH F Þ ½H X ; V B 2
¼ V ðrFX F ðBÞÞ H F V ðrFX BÞ H F V ðrX F ðBÞÞ þ ðH F Þ V ðVX BÞ ¼ V ðrFX F ðBÞÞ V ðF ðrFX BÞÞ V ðF ðrX F ðBÞÞÞ þ V ðF 2 ðrX BÞÞ ¼ V frFX F ðBÞ F ðrFX BÞ F ðrX F ðBÞÞ þ F 2 ðrX BÞg ¼ V fðrFX ÞB ðF ðrX F ÞÞBg; e H F ðH X ; H Y Þ N 2
¼ ½H F H X ; H F H Y H F ½H F H X ; H Y H F ½H X ; H F H Y þ ðH F Þ ½H X ; H Y ¼ H ½FX ; FY ð~c cÞRðFX ; FY Þ H F H ½ðFX Þ; Y H F ð~c cÞRðFX ; Y Þ 2
H F H ½X ; FY H F ð~c cÞRðX ; FY Þ þ ðH F Þ H ½X ; Y þ ðH F Þð~c cÞRðX ; Y Þ ¼ H ðNF ðX ; Y ÞÞ þ ð~c cÞfRðFX ; FY Þ RðFX ; Y Þ RðX ; FY Þ þ RðX ; Y ÞF 2 g: Summing up, we have the following formulas: 8 e H F ðV A; V BÞ ¼ 0; > N > > >
ð6Þ
3. Applications 1. We now suppose that ðF ; gÞ is a K€ ahlerian structure in Mn and r the Riemannian connection determined by the metric g. Then we see that ii(i) F is an almost complex structure in Mn , i.e., F 2 ¼ I; i(ii) rF ¼ 0; (iii) The curvature tensor R of r satisfies RðFX ; FY Þ ¼ RðX ; Y Þ;
8X ; Y 2 T10 ðMn Þ:
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From (iii) we get RðFX ; Y Þ ¼ RðX ; FY Þ; i.e. RðX ; Y Þ hybrid in X and Y . Hence using F 2 ¼ I, we find RðFX ; FY Þ RFX ; Y ÞF RðX ; FY ÞF þ RðX ; Y ÞF 2 ¼ 0: Therefore it follows, from (6) and (ii), that 8 e H F ðV A; V BÞ ¼ 0;
ð7Þ
i.e. curvature tensor R is pure in all arguments [3]. From (i) and (7) we get RðFX ; FY Þ ¼ RðX ; F 2 Y Þ ¼ 0;
ð8Þ
ðRðFX ; Y Þ þ RðX ; FY ÞÞF ¼ 2RðFX ; Y ÞF ¼ 2FRðFX ; Y Þ ¼ 2RðF 2 X ; Y Þ ¼ 0: ð9Þ Thus from (6), (8) and (iii) we have e H F ðV A; V BÞ ¼ 0; N
e H F ðH X ; V BÞ ¼ 0; N
e H F ðH X ; H Y Þ ¼ 0: N
e is zero, since N is skew-symmetric. Thus H F is necessarily Hence, by Lemma N integrable. Summing up, we have
A. Ma gden et al. / Appl. Math. Comput. 156 (2004) 455–461
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Theorem 4. Let ðMn ; gÞ be a B-manifold with tangent structure F 2 ¼ 0 and r the Riemannian connection determined by g. Then the horizontal lift H F of F to Tqp ðMn Þ with respect to r is a tangent structure.
References [1] N. Cengiz, A.A. Salimov, Complete lifts of derivations to tensor bundles, Bol. Soc. Mat. Mexicana 8 (3) (2002) 75–82. [2] A. Ma gden, A.A. Salimov, Horizontal lifts of tensor fields to sections of a tensor bundle, Izv. Vyssh. Uchebu. Zaved. Mat. 45 (3) (2001) 73–76. [3] V.V. Vishnevskii, Affinor structures of affinely connected, Izv. Vyssh. Uchebu. Zaved. Mat 1 (1970) 12–23. [4] A. Ledger, K. Yano, Almost complex structures on tensor bundles, J. Dif. Geom. 1 (1967) 355– 368.