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Diagonal lift in the tensor bundle and its applications
Applied Mathematics and Computation 142 (2003) 309–319 www.elsevier.com/locate/amc
Diagonal lift in the tensor bundle and its applications N. Cengiz, A.A. Salimov
*
Department of Mathematics, Faculty of Arts and Sciences, Atat€urk University, 25240 Erzurum, Turkey
Abstract The purpose of this paper is to define a diagonal lift D g of a Riemannian metric g of a manifold Mn to the tensor bundle Tq1 ðMn Þ of Mn , to associate with D g an Levi–Civita connection of Tq1 ðMn Þ in a natural way and to investigate applications of the diagonal lifts. Ó 2002 Published by Elsevier Science Inc. Keywords: Tensor bundle; Riemannian metric; Diagonal lift; Levi–Civita connection; Killing vector field
1. Introduction Let Mn be an n-dimensional differentiable manifold of class C 1 and Tq1 ðMn Þ the tensor bundle over Mn of tensor of type ð1; qÞ. If xi are local coordinates in a neighborhood U of point x 2 Mn , then a tensor t at x which is an element of Tq1 ðMn Þ is expressible in the form ðxi ; tij1 iq Þ, where tij1 iq are components of t with respect to the natural frame. We may consider ðxi ; tij1 iq Þ ¼ ðxi ; xi Þ ¼ ðxI Þ, q q i ¼ l; . . . ; n, i ¼ n þ 1; . . . ; nð1 þ n Þ, I ¼ 1; . . . ; nðl þ n Þ as local coordinates in a neighborhood p1 ðU Þ (p is the natural projection Tq1 ðMn Þ onto Mn Þ. Let now Mn be a Riemannian manifold with non-degenerate metric g whose components in a coordinate neighborhood U are gji and denote by Chji the Christoffel symbols formed with gji .
*
Corresponding author. E-mail address: [email protected] (A.A. Salimov).
0096-3003/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. doi:10.1016/S0096-3003(02)00305-3
310
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
We denote by Trs ðMn Þ the module over F ðMn Þ (F ðMn Þ is the ring of C 1 functions in Mn ) all tensor fields of class C 1 and of type ðr; sÞ in Mn . Let X 2 T10 ðMn Þ and w 2 T1q ðMn Þ. Then C X 2 T10 ðTq1 ðMn ÞÞ (complete lift) H X 2 T10 ðTq1 ðMn ÞÞ (horizontal lift) and V w 2 T10 ðTq1 ðMn ÞÞ (vertical lift) have, respectively, components [4–6] 0 1 8 ! Xh > 0 > q > C V P > X ¼ @ tm o X k tk w¼ ; m A; > > wkh1 hq h1 hq m h1 mhq ohl X > < l¼1 0 1 ð1Þ Xh > ! > > B C q > H P > > > X ¼ @ X m Ckms ths hq Csmhl thk shq A : 1 1 l¼1
with respect to the natural frame foH g ¼ foh ; oh g, xh ¼ thk1 hq in Tq1 ðMn Þ, where X h and wkh1 hq are respectively local components of X and w. In each coordinate neighborhood U ðxh Þ of Mn , we put o o ¼ dhj h 2 T10 ðMn Þ; j ox ox wj ¼ ol dxj1 dxjq Xj ¼
j ¼ 1; . . . ; n;
j
¼ dkl djh11 dhqq ok dxh1 dxhq 2 T1q ðMn Þ; j ¼ n þ 1; . . . ; nð1 þ nq Þ: Taking account of (1), we easily see that the components of H Xj and V wj are respectively given by 0 1 dhj B q C H Xj ¼ ðAHj Þ ¼ @ P A; Csjhl thk1 shq Ckjs ths 1 hq l¼1
V
wj ¼ ðAHj Þ ¼
dkl djh11
0 j dhqq
with respect to the natural frame foH g where dij -Kronecker delta. We call the set fH Xj ; V wj g the frame adapted to the Riemannian connection r in p1 ðU Þ Tq1 ðMn Þ. On putting Aj ¼ H Xj ;
Aj ¼ V wj ;
we write the adapted frame as fAb g ¼ fAj ; Aj g. It is easily verified that nð1 þ nq Þ local 1-forms A~i ¼ ðA~iH Þ ¼ ðdih ; 0Þ ¼ dxi ;
i ¼ 1; . . . ; n;
ð2Þ
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319 A~i ¼ ðA~iH Þ ¼
Crhs tis1 iq
q X
h
Cshil tir1 siq ; drk dhi11 diqq
l¼1
¼
Crhs tis1 iq
q X
311
! ! h
dxh þ drk dhi11 diqq dxh
Cshil tir1 siq
l¼1
¼ dtir1 iq ;
i ¼ n þ 1; . . . ; nð1 þ nq Þ;
form a coframe fA~a g ¼ fA~i ; A~i g dual to the adapted frame fAb g, i.e., a A~aH AH b ¼ db : 2. Lift D g of a Riemannian g to Tq1 (Mn ) On putting locally D
g ¼ D gji A~j A~i þ D gji A~j A~i
ð3Þ
¼ gji dxj dxi þ glr dj1 i1 djq iq dtjl1 jq dtir1 iq ;
where dji -Kronecker delta, we see that D g defines a tensor field of type ð0; 2Þ in Tq1 ðMn Þ, which called the diagonal lift of the tensor field g to Tq1 ðMn Þ with respect to r. From (3) we prove that D g has components of the form D
D
g ¼ ð gba Þ ¼
gji 0
0 glr dj1 i1 diq iq
ð4Þ
with respect to the adapted frame and components D gji D gji D D g ¼ ð gJI Þ ¼ D ; gji D gji D
gji ¼ gji þ glr d
j1 i 1
d
Cris tis1 iq
jq iq
q X
Cljm tjm1 jq
!
q X
! Cmjjl tjl1 mjq
l¼1
Csiil tir1 siq ;
l¼1
D
gji ¼ glr d
j1 i1
d
jq i q
Cljm tjm1 jq
q X
! Cmjjl tjl1 mjq
l¼1
D
gji ¼ glr dj1 i1 djq iq Cris tis1 iq
q X l¼1
D
gji ¼ glr dj1 i1 djq iq
! Csiil tir1 sjq ;
;
312
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
with respect to the natural frame. The indices a ¼ ði; iÞ, b ¼ ðj; jÞ ¼ 1; . . . ; nð1 þ nq Þ and I ¼ ði; iÞ, J ¼ ðj; jÞ ¼ 1; . . . ; nð1 þ nq Þ indicate the indices with respect to the adapted frame and natural frame, respectively. From (4) it easily follows that if g is a Riemannian metric in Mn , then D g is a Riemannian metric in Tq1 ðMn Þ. Remark 1. The metric D g is similar to that of the Riemannian extension studied by Sasaki [7] in the tangent bundle T01 ðMn Þðq ¼ 0Þ (see also [8, p. 155], for the frame bundle, see [1]). Kowalski [2] studied the Levi–Civita connection of the Sasaki metric on the tangent bundle. Section 3 in this paper will be devoted to study of the Levi–Civita connection of D g in Tq1 ðMn Þ. From (1) and (2) we see that components of C X , H X and V w: C
X a ¼ A~aH C X H ;
H
X a ¼ A~aH H X H ;
V
wa ¼ A~aH V wH
with respect to the adapted frame are given respectively by 0 1 Xi q A; ðC X a Þ ¼ @ t m r X r P t r m i1 iq m i1 miq ril X ðH X a Þ ¼ ðV wa Þ ¼
X
i
l¼1
ð5Þ
;
0
!
0
:
wri1 iq
From (4) and (5) we have D
gðH X ; H Y Þ ¼ gðX ; Y Þ;
ð6Þ
D
gðV w; H Y Þ ¼ 0;
ð7Þ
from (6) we hence have Theorem 1. Let X ; Y 2 T10 ðMn Þ. Then the inner product of the horizontal lifts H X and H Y to Tq1 ðMn Þ with metric D g is equal to the vertical lift of the inner product of X and Y in Mn . From (4) and (5) we have also X D V V gð w; V hÞ ¼ ðgðwðiÞ ; hðiÞ ÞÞ;
ð8Þ
ðiÞ D
gðV w; C Y Þ ¼
X ðiÞ
V
ðgðwðiÞ ; iðrY ÞðiÞ ÞÞ;
ð9Þ
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319 D
gðC X ; C Y Þ ¼ V ðgðX ; Y ÞÞ þ
X
V
ðgðiðrX ÞðiÞ ; iðrY ÞðiÞ ÞÞ;
313
ð10Þ
ðiÞ
where ðiÞ ¼ ði1 iq Þ;
iðrX Þi ¼
m tðiÞ rm X r
q X
! tir1 miq ril X m
or :
l¼1
Since the horizontal (or complete) and the vertical lifts to Tq1 ðMn Þ of vector fields in Mn span the module of vector fields in Tq1 ðMn Þ, formulas (6)–(8) (or (8)– (10)) completely determine the diagonal lift D g of the Riemannian metric g to the tensor bundle Tq1 ðMn Þ. Remark 2. We recall that any element g~ 2 T02 ðT01 ðMn ÞÞ of type ð0; 2Þ in the tangent bundle T01 ðMn Þðq ¼ 0Þ is completely determined by its action an lifts of the type C X1 ; C X2 , where Xi , i ¼ 1; 2 being an arbitrary vector fields in Mn [8, p. 33]. Then D g 2 T02 ðT01 ðMn ÞÞ is completely determined only by (10). 3. Levi–Civita connection of D g We now need components of the non-holonomic object which is important when we use a frame of reference such as fAb g. They are defined by ½Ac ; Ab ¼ Xacb Aa or H ~a Xacb ¼ ðAc AH b A b Ac Þ AH :
According to q X
Aj ¼ oj þ
! Csjhl thk1 shq Ckjs ths 1 hq oh ;
Aj ¼ oj ;
l¼1
the components Xacb are given by Xsij ¼ Xsji ¼
q X
Ctjsl dnr dis11 ditl disqq Cnjr dis11 disqq ;
l¼1
Xsij ¼ Xsji ¼
q X
Rtijsl tsr1 tsq Rijt r tst 1 sq ;
ð11Þ
l¼1
Xsij
¼
Xsij
¼
Xsij
¼ Xsij ¼ Xsij ¼ 0;
where Rhkji are components of the curvature tensor of the Riemannian connection r.
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N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
Components of the Riemannian connection determined by the metric D g are given by 1 1 D a Ccb ¼ D ga ðAc D gb þ Ab D gc A D gcb Þ þ ðXacb þ Xacb þ Xabc Þ; ð12Þ 2 2 where Xacb ¼ D gaD gdb Xdc , D ga are the contravariant components of the metric g with respect to the adapted frame: ji 0 g D ba ð g Þ¼ : ð13Þ 0 glr dj1 i1 djq iq
D
Then, taking account of (11)–(13), we have D
D
Chij ¼ Chij ;
Chij ¼
q X
i
Ctjhl dih11 ditl dhqq dkr ;
l¼1 D
D
D
D
Chij Chij Chij
1 ¼ ghn ghr dm1 j1 dmq jq 2
q X
1 ¼ ghn ghr dm1 i1 dmq iq 2
q X
1 ¼ 2
q X
Rtijhl thk1 thq
! Rniml t tmh 1 tmq
Rhnit tmt 1 mq
l¼1
! Rtnjml tmh 1 tmq
l¼1
;
Rhnjt tmt 1 mq
ð14Þ
;
!
Rkijt tht 1 hq
;
l¼1
j
Chij ¼ Ckil djh11 dhqq ;
D
D
Chij ¼ 0;
Chij ¼ 0:
From (5) and (14) we see that D rVh V w, D rH X H Y , D rVh H Y and D rH X V w have, respectively, components of the form D
D
rVh V w ¼ 0; 0
X i ri Y k
B r HX H Y ¼ @ 1 2
0 D
X iY j
q P
Rtijkl tks1 t...kq
l¼1
B 12 Y j hri1 iq ghn glr dm1 i1 B
d
rVh H Y ¼ B B @
B 12 X i wlj1 jq ghn glr d r HX V w ¼ B @
mq iq
q P l¼1
Yj
q P
!1 Rtnjml tml 1 tmq
Rlnjt tmt 1 mq
Ctjhl hkh1 thq
l¼1
0 D
1 !C 1 A ¼ H ðrX Y Þ þ RðX ; Y Þ; 2 Rsijt tkt 1 kq
m1 j1
d
mq jq
q P
C C C; C A !1
Rtniml tmr 1 tmq Rrnit tmt 1 mq
l¼1
X i ðoi wkh1 hq þ Ckil wlh1 hq Þ
C C; A ð15Þ
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
315
where RðX ; Y Þ has components of the form 0 1 0 ! q RðX ; Y Þ ¼ @ P R t ts i iA s t ijkl k1 ...t...kq Rijt tk1 ...kq X Y l¼1
with respect to the adapted frame (also to the natural frame). Since the horizontal and vertical lifts to Tq1 ðMn Þ span the module of vector field in Tq1 ðMn Þ, formulas (15) completely determine the Riemannian connection D r of the metric D g. We will now define the horizontal lift H r of the Riemannian connection r in Mn to Tq1 ðMn Þ by the conditions H H rVw V h ¼ 0; rVw H Y ¼ 0; ð16Þ H V V H rHX h ¼ ðrX hÞ; rHX H Y ¼ H ðrX Y Þ; for any X ; Y 2 T1q ðMn Þ; w; h 2 T1q ðMn Þ. The horizontal lift ponents H
H
Cims ¼ Cims ;
H i Cm s
H
Cim s ¼ 0; m
¼ Cjsl11 dmi1 1 diq q
q X
Cims ¼ 0;
H
H
r has the com-
Cim s ¼ 0; m
djl11 dmi1 1 Cmsicc diq q ;
c¼1
H i Cms
s
1 ¼ Cjmk ds1 diqq 1 i1
q X
s
djk11 dsi11 Csmic c diqq ;
c¼1 H i Cms
¼ ðom Cjsa1 þ Cjmr1 Crsa Crmc Cjra1 Þtia1 iq þ
q X ðom Casic þ Cjmi1 c Casr þ Crms Caric Þtia1 aiq c¼1
q X ðCjmr1 Casic þ Camic Cjsr1 Þtir1 aiq c¼1
þ H i Cm s
¼0
q q 1XX l r ðC C þ Crmib Clsic Þtij11rliq ; 2 b¼1 c¼1 mic sib
ðxi ¼ tij11iq ; xm ¼ tml11 mq ; xs ¼ tsk11...sq Þ
with respect to the natural frame in Tq1 ðMn Þ. Since the local vector fields H Xi and V wi span the module of vector fields in 1 p ðU Þ Tq1 ðMn Þ, any tensor field of type (1,2) is determined in p1 ðU Þ by its action of H Xi and V wi . We now define a tensor field H S 2 ðT12 ðMn ÞÞ by
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N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
8H H H Sð X ; Y Þ ¼ H ðSðX ; Y ÞÞ; 8X ; Y 2 T1 ðMn Þ; > > < H SðV w; H Y Þ ¼ V ðS ðwÞÞ; 8w 2 T1 ðM Þ;0 Y
H > SðH X ; V hÞ ¼ V ðSX ðhÞÞ; > :H V V Sð w; hÞ ¼ 0;
q
n
8h 2 T1q ðMn Þ;
ð17Þ
where SX ðwÞ, SX ðhÞ 2 T1q ðMn Þ and called H S the horizontal lift of S 2 T12 ðMn Þ to T1q ðMn Þ. Denoting by T and Te , respectively, the torsion tensors of r and H r. Directly from the definition of the torsion tensor, we get 8 < Te ðV w; V hÞ ¼ H rVw V h H rVh V w ½V w; V h; H H H V V H e V H Y ; : Te ðHw; HY Þ ¼ HrVw HY HrHY Hw ½ Hw; H T ð X ; Y Þ ¼ rHX Y rHY X ½ X ; Y : On other hand, let R denote the curvature tensor field of the connection V. Then [3] ( ½V w; V h ¼ 0; ½V w; H Y ¼ V ðrY wÞ; ð18Þ ½H X ; H Y ¼ H ½X ; Y þ RðX ; Y Þ: Taking account of (16)–(18), we obtain Te ðV w; V hÞ ¼ 0; Te ðV w; H Y Þ ¼ 0; Te ðH X ; H Y Þ ¼ H T ðH X ; H Y Þ RðX ; Y Þ ¼ H ðT ðX ; Y ÞÞ RðX ; Y Þ ¼ RðX ; Y Þ: Therefore we have Theorem 2. When r is a Riemannian connection, H r is torsionless if r is locally flat, i.e. T ¼ 0 and R ¼ 0. We put D
g~ ¼ g~ji Aj Ai þ dj1 i1 djq iq g~lr Aj Ai
in T1q ðMn Þ. From (16), we have 8H h r HX Ai ¼ X s C > > si Ah ; > Pq > > rVw Ai ¼ 0; : H rVw Ai ¼ 0;
ð19Þ
ð20Þ
for any X 2 T10 ðMn Þ, w 2 T1q ðMn Þ. Thus, according to (19) and (20), we obtain
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
H H
rHX D g~ ¼ D ðrX g~Þ; rVw D g~ ¼ 0:
317
ð21Þ
Let rX g ¼ 0, then rX g~ ¼ 0. Thus, taking account (21), rX g ¼ 0 and gac D g~cb ¼ dba , we obtain H rHX D g ¼ 0; H rVw D g ¼ 0:
D
Thus we have Theorem 3. Let Mn be a Riemannian manifold with metric g. Then the horizontal lift H r of the Riemannian connection r is a metric connection with respect to D g.
4. Killing vector fields A vector field X 2 T10 ðMn Þ is said to be an infinitesimal isometry or a Killing vector field of a Riemannian manifold with metric g, if LX g ¼ 0 [9, p. 178]. In terms of components gji of g, X is a Killing vector field if and only if LX gji ¼ X a ra gji þ gai rj X a þ gja ri X a ¼ rj Xi þ ri Xj ¼ 0; X a being components of X, where r is the Riemannian connection of the metric g. The associated covector fields of the complete and horizontal lifts to Tq1 ðMn Þ with the metric D g are given respectively by !! q X j1 i1 jq iq C a C D m r r m ð Xb Þ ¼ ð gba X Þ ¼ Xj ; glr d d ti1 iq rm X ti1 miq riu X ; l¼1 H
H
D
a
ð Xb Þ ¼ ð gba X Þ ¼ ðXj ; 0Þ ð22Þ
with respect to the adapted frame. We now compute the Lie derivatives of the metric D g with respect to C X and H X . The Lie derivatives of D g with respect to C X and H X have respectively components 1 C Aðb XaÞ þ D CcðbaÞ C Xc r b C X a þ D r a C Xb ¼ 2 D D LCX gji LCX gji ¼ ; LCX D gji LCX D gji
ðLCX D gba Þ ¼
D
318
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
8 LCX D gji ¼ rj Xi þ ri Xj > > > > > LCX D gji ¼ ðrj rt Xr þ Rnjtr X n Þtpt 1 pq di1 p1 diq pq > > Pq > i1 p1 il t iq pq > t t n s > > < l¼1 ½ðrj rpl X þ Rnjpl X Þgsr tp1 tpq d d d LCX D gji ¼ ðri rt Xl þ Rnitl X n Þtpt 1 pq dj1 p1 djq pq > > j1 p 1 > Pq ½ðri rp X t þ Rnip t X n Þgsl ts djl t djq pq > l l p1 tpq d l¼1 > > P > > > LCX D gji ¼ ðrl Xr þ rr Xl Þdj1 i1 djq iq glr ql¼1 ðrm X jl dj1 i1 dmil djq iq > > : þrm X il dj1 i1 djl m djq iq Þ; ð23Þ
1 ðLHX D gba Þ ¼ ðD rb H Xa þ D ra H Xb Þ ¼ ðAðb H XaÞ þ D CcðbaÞ H Xc Þ 2 D D LHX gji LHX gji ¼ LHX D gji LHX D gji 8 LHX D gji ¼ rj Xi þ ri Xj ; > > > > LHX D gji ¼ Rhjtr Xh tmt 1 mq dm1 i1 dmq iq ; > > > < Pq Rnjm t X n ghr th dm1 i1 dmq iq ; l¼1
m1 tmq
l
LHX D gji ¼ Rhitl Xh tmt 1 mq dm1 j1 dmq jq ; > > > Pq > m1 j1 t n h > dmq jq ; > > l¼1 Rniml X ghl tm1 tmq d : D LHX gji ¼ 0 with respect to the adapted frame in Tq1 ðMn Þ. Since we have ri rk Xt þ Rlikt X l ¼ 0 as a consequence of LX gji ¼ rj Xi þ ri Xj ¼ 0 (see [9, p.17]), we conclude by means of (23) that the complete lift C X is a Killing vector field in Tq1 ðMn Þ if X is a Killing vector field in Tq1 ðMn Þ. We have Rhjtr Xh ¼ 0 and
Rhjm t X h ¼ 0
as a consequence of the vanishing of the second covariant derivative of X. Summing up these results, we have Theorem 4. Sufficient conditions in order that (a) the complete, (b) horizontal lifts to Tq1 ðMn Þ with metric D g of, a vector field X in Mn , be a Killing vector field in Tq1 ðMn Þ respectively are that, (a) X is a Killing vector field with vanishing covariant derivative in Mn , (b) X is a Killing vector field with vanishing second covariant derivative in Mn .
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319
References [1] L.A. Cordero, C.T.J. Dotson, M. de Leon, Differential Geometry of Frame Bundles, Kluwer Academic Publishers, Dordrecht, 1989. [2] O. Kowalski, Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold, J. Reine Angew. Math. 250 (1971) 124–129. [3] A. Ledger, K. Yano, Almost complex structures on tensor bundles, J. Differ. Geom. 1 (1976) 355–368. [4] A.A. Salimov, A new method in theory of lifts of tensor fields to a tensor bundle, IZV. Vuz. Math. 3 (1994) 69–75. [5] A.A. Salimov, The generalized Yano–Ako operator and the complete lift of tensor fields, Tensor N.S. Jpn. 55 (2) (1994) 142–146. [6] A.A. Salimov, A. Magden, Complete lifts of tensor fields on a pure cross-section in the tensor bundle Tq1 ðMnÞ, Note Di Matematica 18 (1) (1998) 27–37. [7] S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 (1958) 338–354. [8] K. Yano, S. Ishihara, Tangent and Cotangent Bundles, Mareel Dekker, New York, 1973. [9] K. Yano, The Theory of Lie Derivatives and Its Applications, Elsevier, Amsterdam, 1957.
Diagonal lift in the tensor bundle and its applications N. Cengiz, A.A. Salimov
*
Department of Mathematics, Faculty of Arts and Sciences, Atat€urk University, 25240 Erzurum, Turkey
Abstract The purpose of this paper is to define a diagonal lift D g of a Riemannian metric g of a manifold Mn to the tensor bundle Tq1 ðMn Þ of Mn , to associate with D g an Levi–Civita connection of Tq1 ðMn Þ in a natural way and to investigate applications of the diagonal lifts. Ó 2002 Published by Elsevier Science Inc. Keywords: Tensor bundle; Riemannian metric; Diagonal lift; Levi–Civita connection; Killing vector field
1. Introduction Let Mn be an n-dimensional differentiable manifold of class C 1 and Tq1 ðMn Þ the tensor bundle over Mn of tensor of type ð1; qÞ. If xi are local coordinates in a neighborhood U of point x 2 Mn , then a tensor t at x which is an element of Tq1 ðMn Þ is expressible in the form ðxi ; tij1 iq Þ, where tij1 iq are components of t with respect to the natural frame. We may consider ðxi ; tij1 iq Þ ¼ ðxi ; xi Þ ¼ ðxI Þ, q q i ¼ l; . . . ; n, i ¼ n þ 1; . . . ; nð1 þ n Þ, I ¼ 1; . . . ; nðl þ n Þ as local coordinates in a neighborhood p1 ðU Þ (p is the natural projection Tq1 ðMn Þ onto Mn Þ. Let now Mn be a Riemannian manifold with non-degenerate metric g whose components in a coordinate neighborhood U are gji and denote by Chji the Christoffel symbols formed with gji .
*
Corresponding author. E-mail address: [email protected] (A.A. Salimov).
0096-3003/02/$ - see front matter Ó 2002 Published by Elsevier Science Inc. doi:10.1016/S0096-3003(02)00305-3
310
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
We denote by Trs ðMn Þ the module over F ðMn Þ (F ðMn Þ is the ring of C 1 functions in Mn ) all tensor fields of class C 1 and of type ðr; sÞ in Mn . Let X 2 T10 ðMn Þ and w 2 T1q ðMn Þ. Then C X 2 T10 ðTq1 ðMn ÞÞ (complete lift) H X 2 T10 ðTq1 ðMn ÞÞ (horizontal lift) and V w 2 T10 ðTq1 ðMn ÞÞ (vertical lift) have, respectively, components [4–6] 0 1 8 ! Xh > 0 > q > C V P > X ¼ @ tm o X k tk w¼ ; m A; > > wkh1 hq h1 hq m h1 mhq ohl X > < l¼1 0 1 ð1Þ Xh > ! > > B C q > H P > > > X ¼ @ X m Ckms ths hq Csmhl thk shq A : 1 1 l¼1
with respect to the natural frame foH g ¼ foh ; oh g, xh ¼ thk1 hq in Tq1 ðMn Þ, where X h and wkh1 hq are respectively local components of X and w. In each coordinate neighborhood U ðxh Þ of Mn , we put o o ¼ dhj h 2 T10 ðMn Þ; j ox ox wj ¼ ol dxj1 dxjq Xj ¼
j ¼ 1; . . . ; n;
j
¼ dkl djh11 dhqq ok dxh1 dxhq 2 T1q ðMn Þ; j ¼ n þ 1; . . . ; nð1 þ nq Þ: Taking account of (1), we easily see that the components of H Xj and V wj are respectively given by 0 1 dhj B q C H Xj ¼ ðAHj Þ ¼ @ P A; Csjhl thk1 shq Ckjs ths 1 hq l¼1
V
wj ¼ ðAHj Þ ¼
dkl djh11
0 j dhqq
with respect to the natural frame foH g where dij -Kronecker delta. We call the set fH Xj ; V wj g the frame adapted to the Riemannian connection r in p1 ðU Þ Tq1 ðMn Þ. On putting Aj ¼ H Xj ;
Aj ¼ V wj ;
we write the adapted frame as fAb g ¼ fAj ; Aj g. It is easily verified that nð1 þ nq Þ local 1-forms A~i ¼ ðA~iH Þ ¼ ðdih ; 0Þ ¼ dxi ;
i ¼ 1; . . . ; n;
ð2Þ
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319 A~i ¼ ðA~iH Þ ¼
Crhs tis1 iq
q X
h
Cshil tir1 siq ; drk dhi11 diqq
l¼1
¼
Crhs tis1 iq
q X
311
! ! h
dxh þ drk dhi11 diqq dxh
Cshil tir1 siq
l¼1
¼ dtir1 iq ;
i ¼ n þ 1; . . . ; nð1 þ nq Þ;
form a coframe fA~a g ¼ fA~i ; A~i g dual to the adapted frame fAb g, i.e., a A~aH AH b ¼ db : 2. Lift D g of a Riemannian g to Tq1 (Mn ) On putting locally D
g ¼ D gji A~j A~i þ D gji A~j A~i
ð3Þ
¼ gji dxj dxi þ glr dj1 i1 djq iq dtjl1 jq dtir1 iq ;
where dji -Kronecker delta, we see that D g defines a tensor field of type ð0; 2Þ in Tq1 ðMn Þ, which called the diagonal lift of the tensor field g to Tq1 ðMn Þ with respect to r. From (3) we prove that D g has components of the form D
D
g ¼ ð gba Þ ¼
gji 0
0 glr dj1 i1 diq iq
ð4Þ
with respect to the adapted frame and components D gji D gji D D g ¼ ð gJI Þ ¼ D ; gji D gji D
gji ¼ gji þ glr d
j1 i 1
d
Cris tis1 iq
jq iq
q X
Cljm tjm1 jq
!
q X
! Cmjjl tjl1 mjq
l¼1
Csiil tir1 siq ;
l¼1
D
gji ¼ glr d
j1 i1
d
jq i q
Cljm tjm1 jq
q X
! Cmjjl tjl1 mjq
l¼1
D
gji ¼ glr dj1 i1 djq iq Cris tis1 iq
q X l¼1
D
gji ¼ glr dj1 i1 djq iq
! Csiil tir1 sjq ;
;
312
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
with respect to the natural frame. The indices a ¼ ði; iÞ, b ¼ ðj; jÞ ¼ 1; . . . ; nð1 þ nq Þ and I ¼ ði; iÞ, J ¼ ðj; jÞ ¼ 1; . . . ; nð1 þ nq Þ indicate the indices with respect to the adapted frame and natural frame, respectively. From (4) it easily follows that if g is a Riemannian metric in Mn , then D g is a Riemannian metric in Tq1 ðMn Þ. Remark 1. The metric D g is similar to that of the Riemannian extension studied by Sasaki [7] in the tangent bundle T01 ðMn Þðq ¼ 0Þ (see also [8, p. 155], for the frame bundle, see [1]). Kowalski [2] studied the Levi–Civita connection of the Sasaki metric on the tangent bundle. Section 3 in this paper will be devoted to study of the Levi–Civita connection of D g in Tq1 ðMn Þ. From (1) and (2) we see that components of C X , H X and V w: C
X a ¼ A~aH C X H ;
H
X a ¼ A~aH H X H ;
V
wa ¼ A~aH V wH
with respect to the adapted frame are given respectively by 0 1 Xi q A; ðC X a Þ ¼ @ t m r X r P t r m i1 iq m i1 miq ril X ðH X a Þ ¼ ðV wa Þ ¼
X
i
l¼1
ð5Þ
;
0
!
0
:
wri1 iq
From (4) and (5) we have D
gðH X ; H Y Þ ¼ gðX ; Y Þ;
ð6Þ
D
gðV w; H Y Þ ¼ 0;
ð7Þ
from (6) we hence have Theorem 1. Let X ; Y 2 T10 ðMn Þ. Then the inner product of the horizontal lifts H X and H Y to Tq1 ðMn Þ with metric D g is equal to the vertical lift of the inner product of X and Y in Mn . From (4) and (5) we have also X D V V gð w; V hÞ ¼ ðgðwðiÞ ; hðiÞ ÞÞ;
ð8Þ
ðiÞ D
gðV w; C Y Þ ¼
X ðiÞ
V
ðgðwðiÞ ; iðrY ÞðiÞ ÞÞ;
ð9Þ
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319 D
gðC X ; C Y Þ ¼ V ðgðX ; Y ÞÞ þ
X
V
ðgðiðrX ÞðiÞ ; iðrY ÞðiÞ ÞÞ;
313
ð10Þ
ðiÞ
where ðiÞ ¼ ði1 iq Þ;
iðrX Þi ¼
m tðiÞ rm X r
q X
! tir1 miq ril X m
or :
l¼1
Since the horizontal (or complete) and the vertical lifts to Tq1 ðMn Þ of vector fields in Mn span the module of vector fields in Tq1 ðMn Þ, formulas (6)–(8) (or (8)– (10)) completely determine the diagonal lift D g of the Riemannian metric g to the tensor bundle Tq1 ðMn Þ. Remark 2. We recall that any element g~ 2 T02 ðT01 ðMn ÞÞ of type ð0; 2Þ in the tangent bundle T01 ðMn Þðq ¼ 0Þ is completely determined by its action an lifts of the type C X1 ; C X2 , where Xi , i ¼ 1; 2 being an arbitrary vector fields in Mn [8, p. 33]. Then D g 2 T02 ðT01 ðMn ÞÞ is completely determined only by (10). 3. Levi–Civita connection of D g We now need components of the non-holonomic object which is important when we use a frame of reference such as fAb g. They are defined by ½Ac ; Ab ¼ Xacb Aa or H ~a Xacb ¼ ðAc AH b A b Ac Þ AH :
According to q X
Aj ¼ oj þ
! Csjhl thk1 shq Ckjs ths 1 hq oh ;
Aj ¼ oj ;
l¼1
the components Xacb are given by Xsij ¼ Xsji ¼
q X
Ctjsl dnr dis11 ditl disqq Cnjr dis11 disqq ;
l¼1
Xsij ¼ Xsji ¼
q X
Rtijsl tsr1 tsq Rijt r tst 1 sq ;
ð11Þ
l¼1
Xsij
¼
Xsij
¼
Xsij
¼ Xsij ¼ Xsij ¼ 0;
where Rhkji are components of the curvature tensor of the Riemannian connection r.
314
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
Components of the Riemannian connection determined by the metric D g are given by 1 1 D a Ccb ¼ D ga ðAc D gb þ Ab D gc A D gcb Þ þ ðXacb þ Xacb þ Xabc Þ; ð12Þ 2 2 where Xacb ¼ D gaD gdb Xdc , D ga are the contravariant components of the metric g with respect to the adapted frame: ji 0 g D ba ð g Þ¼ : ð13Þ 0 glr dj1 i1 djq iq
D
Then, taking account of (11)–(13), we have D
D
Chij ¼ Chij ;
Chij ¼
q X
i
Ctjhl dih11 ditl dhqq dkr ;
l¼1 D
D
D
D
Chij Chij Chij
1 ¼ ghn ghr dm1 j1 dmq jq 2
q X
1 ¼ ghn ghr dm1 i1 dmq iq 2
q X
1 ¼ 2
q X
Rtijhl thk1 thq
! Rniml t tmh 1 tmq
Rhnit tmt 1 mq
l¼1
! Rtnjml tmh 1 tmq
l¼1
;
Rhnjt tmt 1 mq
ð14Þ
;
!
Rkijt tht 1 hq
;
l¼1
j
Chij ¼ Ckil djh11 dhqq ;
D
D
Chij ¼ 0;
Chij ¼ 0:
From (5) and (14) we see that D rVh V w, D rH X H Y , D rVh H Y and D rH X V w have, respectively, components of the form D
D
rVh V w ¼ 0; 0
X i ri Y k
B r HX H Y ¼ @ 1 2
0 D
X iY j
q P
Rtijkl tks1 t...kq
l¼1
B 12 Y j hri1 iq ghn glr dm1 i1 B
d
rVh H Y ¼ B B @
B 12 X i wlj1 jq ghn glr d r HX V w ¼ B @
mq iq
q P l¼1
Yj
q P
!1 Rtnjml tml 1 tmq
Rlnjt tmt 1 mq
Ctjhl hkh1 thq
l¼1
0 D
1 !C 1 A ¼ H ðrX Y Þ þ RðX ; Y Þ; 2 Rsijt tkt 1 kq
m1 j1
d
mq jq
q P
C C C; C A !1
Rtniml tmr 1 tmq Rrnit tmt 1 mq
l¼1
X i ðoi wkh1 hq þ Ckil wlh1 hq Þ
C C; A ð15Þ
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
315
where RðX ; Y Þ has components of the form 0 1 0 ! q RðX ; Y Þ ¼ @ P R t ts i iA s t ijkl k1 ...t...kq Rijt tk1 ...kq X Y l¼1
with respect to the adapted frame (also to the natural frame). Since the horizontal and vertical lifts to Tq1 ðMn Þ span the module of vector field in Tq1 ðMn Þ, formulas (15) completely determine the Riemannian connection D r of the metric D g. We will now define the horizontal lift H r of the Riemannian connection r in Mn to Tq1 ðMn Þ by the conditions H H rVw V h ¼ 0; rVw H Y ¼ 0; ð16Þ H V V H rHX h ¼ ðrX hÞ; rHX H Y ¼ H ðrX Y Þ; for any X ; Y 2 T1q ðMn Þ; w; h 2 T1q ðMn Þ. The horizontal lift ponents H
H
Cims ¼ Cims ;
H i Cm s
H
Cim s ¼ 0; m
¼ Cjsl11 dmi1 1 diq q
q X
Cims ¼ 0;
H
H
r has the com-
Cim s ¼ 0; m
djl11 dmi1 1 Cmsicc diq q ;
c¼1
H i Cms
s
1 ¼ Cjmk ds1 diqq 1 i1
q X
s
djk11 dsi11 Csmic c diqq ;
c¼1 H i Cms
¼ ðom Cjsa1 þ Cjmr1 Crsa Crmc Cjra1 Þtia1 iq þ
q X ðom Casic þ Cjmi1 c Casr þ Crms Caric Þtia1 aiq c¼1
q X ðCjmr1 Casic þ Camic Cjsr1 Þtir1 aiq c¼1
þ H i Cm s
¼0
q q 1XX l r ðC C þ Crmib Clsic Þtij11rliq ; 2 b¼1 c¼1 mic sib
ðxi ¼ tij11iq ; xm ¼ tml11 mq ; xs ¼ tsk11...sq Þ
with respect to the natural frame in Tq1 ðMn Þ. Since the local vector fields H Xi and V wi span the module of vector fields in 1 p ðU Þ Tq1 ðMn Þ, any tensor field of type (1,2) is determined in p1 ðU Þ by its action of H Xi and V wi . We now define a tensor field H S 2 ðT12 ðMn ÞÞ by
316
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
8H H H Sð X ; Y Þ ¼ H ðSðX ; Y ÞÞ; 8X ; Y 2 T1 ðMn Þ; > > < H SðV w; H Y Þ ¼ V ðS ðwÞÞ; 8w 2 T1 ðM Þ;0 Y
H > SðH X ; V hÞ ¼ V ðSX ðhÞÞ; > :H V V Sð w; hÞ ¼ 0;
q
n
8h 2 T1q ðMn Þ;
ð17Þ
where SX ðwÞ, SX ðhÞ 2 T1q ðMn Þ and called H S the horizontal lift of S 2 T12 ðMn Þ to T1q ðMn Þ. Denoting by T and Te , respectively, the torsion tensors of r and H r. Directly from the definition of the torsion tensor, we get 8 < Te ðV w; V hÞ ¼ H rVw V h H rVh V w ½V w; V h; H H H V V H e V H Y ; : Te ðHw; HY Þ ¼ HrVw HY HrHY Hw ½ Hw; H T ð X ; Y Þ ¼ rHX Y rHY X ½ X ; Y : On other hand, let R denote the curvature tensor field of the connection V. Then [3] ( ½V w; V h ¼ 0; ½V w; H Y ¼ V ðrY wÞ; ð18Þ ½H X ; H Y ¼ H ½X ; Y þ RðX ; Y Þ: Taking account of (16)–(18), we obtain Te ðV w; V hÞ ¼ 0; Te ðV w; H Y Þ ¼ 0; Te ðH X ; H Y Þ ¼ H T ðH X ; H Y Þ RðX ; Y Þ ¼ H ðT ðX ; Y ÞÞ RðX ; Y Þ ¼ RðX ; Y Þ: Therefore we have Theorem 2. When r is a Riemannian connection, H r is torsionless if r is locally flat, i.e. T ¼ 0 and R ¼ 0. We put D
g~ ¼ g~ji Aj Ai þ dj1 i1 djq iq g~lr Aj Ai
in T1q ðMn Þ. From (16), we have 8H h r HX Ai ¼ X s C > > si Ah ; > Pq
ð19Þ
ð20Þ
for any X 2 T10 ðMn Þ, w 2 T1q ðMn Þ. Thus, according to (19) and (20), we obtain
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
H H
rHX D g~ ¼ D ðrX g~Þ; rVw D g~ ¼ 0:
317
ð21Þ
Let rX g ¼ 0, then rX g~ ¼ 0. Thus, taking account (21), rX g ¼ 0 and gac D g~cb ¼ dba , we obtain H rHX D g ¼ 0; H rVw D g ¼ 0:
D
Thus we have Theorem 3. Let Mn be a Riemannian manifold with metric g. Then the horizontal lift H r of the Riemannian connection r is a metric connection with respect to D g.
4. Killing vector fields A vector field X 2 T10 ðMn Þ is said to be an infinitesimal isometry or a Killing vector field of a Riemannian manifold with metric g, if LX g ¼ 0 [9, p. 178]. In terms of components gji of g, X is a Killing vector field if and only if LX gji ¼ X a ra gji þ gai rj X a þ gja ri X a ¼ rj Xi þ ri Xj ¼ 0; X a being components of X, where r is the Riemannian connection of the metric g. The associated covector fields of the complete and horizontal lifts to Tq1 ðMn Þ with the metric D g are given respectively by !! q X j1 i1 jq iq C a C D m r r m ð Xb Þ ¼ ð gba X Þ ¼ Xj ; glr d d ti1 iq rm X ti1 miq riu X ; l¼1 H
H
D
a
ð Xb Þ ¼ ð gba X Þ ¼ ðXj ; 0Þ ð22Þ
with respect to the adapted frame. We now compute the Lie derivatives of the metric D g with respect to C X and H X . The Lie derivatives of D g with respect to C X and H X have respectively components 1 C Aðb XaÞ þ D CcðbaÞ C Xc r b C X a þ D r a C Xb ¼ 2 D D LCX gji LCX gji ¼ ; LCX D gji LCX D gji
ðLCX D gba Þ ¼
D
318
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
8 LCX D gji ¼ rj Xi þ ri Xj > > > > > LCX D gji ¼ ðrj rt Xr þ Rnjtr X n Þtpt 1 pq di1 p1 diq pq > > Pq > i1 p1 il t iq pq > t t n s > > < l¼1 ½ðrj rpl X þ Rnjpl X Þgsr tp1 tpq d d d LCX D gji ¼ ðri rt Xl þ Rnitl X n Þtpt 1 pq dj1 p1 djq pq > > j1 p 1 > Pq ½ðri rp X t þ Rnip t X n Þgsl ts djl t djq pq > l l p1 tpq d l¼1 > > P > > > LCX D gji ¼ ðrl Xr þ rr Xl Þdj1 i1 djq iq glr ql¼1 ðrm X jl dj1 i1 dmil djq iq > > : þrm X il dj1 i1 djl m djq iq Þ; ð23Þ
1 ðLHX D gba Þ ¼ ðD rb H Xa þ D ra H Xb Þ ¼ ðAðb H XaÞ þ D CcðbaÞ H Xc Þ 2 D D LHX gji LHX gji ¼ LHX D gji LHX D gji 8 LHX D gji ¼ rj Xi þ ri Xj ; > > > > LHX D gji ¼ Rhjtr Xh tmt 1 mq dm1 i1 dmq iq ; > > > < Pq Rnjm t X n ghr th dm1 i1 dmq iq ; l¼1
m1 tmq
l
LHX D gji ¼ Rhitl Xh tmt 1 mq dm1 j1 dmq jq ; > > > Pq > m1 j1 t n h > dmq jq ; > > l¼1 Rniml X ghl tm1 tmq d : D LHX gji ¼ 0 with respect to the adapted frame in Tq1 ðMn Þ. Since we have ri rk Xt þ Rlikt X l ¼ 0 as a consequence of LX gji ¼ rj Xi þ ri Xj ¼ 0 (see [9, p.17]), we conclude by means of (23) that the complete lift C X is a Killing vector field in Tq1 ðMn Þ if X is a Killing vector field in Tq1 ðMn Þ. We have Rhjtr Xh ¼ 0 and
Rhjm t X h ¼ 0
as a consequence of the vanishing of the second covariant derivative of X. Summing up these results, we have Theorem 4. Sufficient conditions in order that (a) the complete, (b) horizontal lifts to Tq1 ðMn Þ with metric D g of, a vector field X in Mn , be a Killing vector field in Tq1 ðMn Þ respectively are that, (a) X is a Killing vector field with vanishing covariant derivative in Mn , (b) X is a Killing vector field with vanishing second covariant derivative in Mn .
N. Cengiz, A.A. Salimov / Appl. Math. Comput. 142 (2003) 309–319
319
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