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PureMathematicsnØêÆ,2022,12(10),1649-1654
PublishedOnlineOctober2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1210178
V-Kenmotsu6/þCYamabe
áf
¸¸¸ÚÚÚ999
∗
§§§444ïï襤¤
†
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2022c913F¶¹^Fϵ2022c1012F¶uÙFϵ2022c1020F
Á‡
|^Lie êŽf,C‡©Žf±9/• þ|5Ÿ,y²3äkV-Kenmotsu (C
Yamabe áf¥,XJ•31w¼êf,¦ƒ>1−/ªηØC,KÙ³•þ|´Killing •þ
|"
'…c
V-Kenmotsu6/§/•þ|§CYamabe áf§Killing •þ|
AlmostYamabeSolitonsonHyperbolic
KenmotsuManifolds
HelongHan
∗
,JianchengLiu
†
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Sep.13
th
,2022;accepted:Oct.12
th
,2022;published:Oct.20
th
,2022
∗1˜Šö"
†ÏÕŠö"
©ÙÚ^:¸Ú9,4ï¤.V-Kenmotsu6/þCYamabeáf[J].nØêÆ,2022,12(10):1649-1654.
DOI:10.12677/pm.2022.1210178
¸Ú9§4ï¤
Abstract
ByusingthepropertiesofLie-derivativeoperator,covariantderivativeoperatorand
conformalvectorfield,we prove thatin almostYamabesolitonswithhyperbolic Ken-
motsustructrue,ifthereexistsasmoothfunctionfthatleavesthecontact1-formη
invariant,thenitspotentialvectorfieldsareKillingvectorfields.
Keywords
HyperbolicKenmotsuManifold,ConformalVectorField,AlmostYamabeSoliton,
KillingVectorField
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó9̇(J
(M
n
,g) ´n‘–iù6/,PX(M)´Mþ1wƒ•þ|8Ü.XJ• 3V∈X(M)
Ú˜‡~êλ,¦
1
2
L
V
g= (λ−r)g,
KM
n
¡•˜‡Yamabe áf, džV¡•Yamabeáf³•þ|,λ¡•áf~ê, L
V
g
L«Ýþg÷³•þ|V••Lieê, rL«M
n
êþ-Ç.3λ>0(=0,<0)ž,K¡
Yamabe áf´Â (-½,*Ü).
3ØÓ-Ç^‡e,Yamabeáf®²NõÆö2•ïÄ([1],[2],[3],[4]),AO/,Hsu
3[5]¥y²?¿;—Yamabeáfêþ-ǘ½´~ê.Yamabeáfí2C
Yamabe áf3©z[6] ¥Barbosa-Ribeiro 0Xe:
½Â1(M,g) ´˜‡–iù6/,XJ•3Mþƒ•þ|VÚ1w¼êλ,¦
1
2
L
V
g= (λ−r)g,(1)
K¡M•˜‡CYamabeáf,P•(g,V,λ).w,,λ•~êž,CYamabeáf=•
DOI:10.12677/pm.2022.12101781650nØêÆ
¸Ú9§4ï¤
Yamabe áf.Ó/,λ>0(= 0,<0) ž,(g,V,λ) ¡•´Â (-½,*Ü).
,˜•¡,UpadhyayÚDube[7] JÑCƒ>V-(Vg,ÓžÙ¦Æö?˜Úï
Ä.31972c,Kenmotsu[8]ïÄ÷v˜AÏ^‡ƒ>iù6/©a,¿òù«6/·
¶•Kenmotsu6/. Kenmotsuy²Kenmotsu6/ÛÜþ´˜‡«mIÚ˜‡äkòȼ
êf(t)=se
t
K¨ahler6/M¤/¤òÈI×
f
M,s´š"~ê.3[9]¥,GhoshïÄ
Kenmotsu 6/¥CRicci áfÚCYamabe áf,Äud,©?ØV-Kenmotsu 6/þ
CYamabe áf³•þ|5Ÿ,e¡Ì‡½n.
½n1(M
2n+1
,g,V,λ) ´(2n+1) ‘äkV-Kenmotsu (˜‡CYamabe áf,
XJ•31w¼êf,¦ƒ>1−/ªηØC,=L
V
η= fη,KV´Killing •þ|.
©ëì©z[9] ‰ÑV-Kenmotsu 6/þCYamabe áfƒ'5Ÿ.Ù(Œ—
©•n‡Ü©.Äk0˜'uCYamabe áfïÄyG¿‰Ñ©̇½n,Ùg
0V-Kenmotsu 6/Vg9Ù5Ÿ,•©¥½ny²‰O,•K‰Ñ©½n
y²L§.
2.ý•£9Ún
M
2n+1
´˜‡2n+ 1 ‘1w6/,XJ•3˜‡(1,1).Üþ|φ,˜‡ƒ•þ|ζÚƒ
>1−/ªη¦
φ
2
= I+η⊗ζ,η(ζ) = −1,(2)
Ù ¥I´M
2n+1
ƒ mþðgÓ,⊗“LÜþÈ,K¡M
2n+1
äk Cƒ>V-(,P•
(φ,ζ,η).džM
2n+1
\þ˜‡Cƒ>V-(Ò¡•˜‡Cƒ>V-6/,P•(M
n+1
,φ,ζ,η).
XJ˜‡Cƒ>V-6/M
2n+1
þ•3–iùÝþg÷v
g(φX,φY) = −g(X,Y)−η(X)η(Y),∀X,Y∈X(M),
K¡M
2n+1
´Cƒ>V-–Ýþ6/,(φ,ζ,η,g)¡•Cƒ>V-–Ýþ(.•?˜Úé
∀X,Y∈X(M) k
(∇
X
φ)Y= g(φX,Y)ζ−η(Y)φ(X)
¤á,K¡Ù•V-Kenmotsu 6/,džk±eª¤á([10])
∇
X
ζ= −X−η(X)ζ,(3)
(∇
X
η)Y= g(φX,φY) = −g(X,Y)−η(X)η(Y),
R(X,Y)ζ= η(Y)X−η(X)Y,
R(X,ζ)ζ= −X−η(X)ζ,(4)
R(ζ,X)Y= g(X,Y)ζ−η(Y)X,
DOI:10.12677/pm.2022.12101781651nØêÆ
¸Ú9§4ï¤
S(X,ζ) = 2nη(X),S(ζ,ζ) = −2n,Qζ= 2nζ.(5)
Ù¥R,SÚQ©O´M
2n+1
þ-ÇÜþ,Ricci ÜþÚRicci Žf,…S(X,Y) = g(QX,Y).
½Â2–iù6/(M
n
,g) þ•þ|VXJ÷v
L
V
g= 2ρg,
K¡V´˜‡/•þ|,Ù¥ρ´Mþ1w¼ê,¡•V/Ïf.XJρ=0,KV´
Killing•þ|.w,3˜‡CYamabe áf¥,V´±ρ= (r−λ)•/Ïf/•þ|.
,•y²©(Ø,·‚I‡e¡Ún(„[11]).
Ún1é32n+1 ‘iù6/Mþ˜‡/•þ|V,keúª¤á
(L
V
R)(X,Y)Z=g(∇
X
Dρ,Z)Y−g(∇
Y
Dρ,Z)X
+g(X,Z)∇
Y
Dρ−g(Y,Z)∇
X
Dρ,
(6)
(L
V
S)(X,Y) = (1−2n)g(∇
X
Dρ,Y)+(∆ρ)g(X,Y),(7)
L
V
r= −2rρ+4n∆ρ,
Ù¥X,Y, Z´Mþ?¿•þ|,…∆ = −divD´'uÝþgLaplacian Žf, div´ÑÝŽ
f,D´FÝŽf.
3.̇½ny²
½n1y²Šâ•§(1)Ú½Â1Œ±`³•þ|V´/,Ïdρ=(λ−r).é
η(ζ) = g(ζ,ζ) = −1 ¦÷V••Lie ê,^•§(1) Ú(2)
(L
V
η)ζ= −η(L
V
ζ) = −ρ.
džb³•þ|V¦ƒ>/ηØC,=L
V
η= fη,f∈C
∞
(M),Kk(fη)ζ= −ρ,f= ρ.
u´
L
V
η= ρη,L
V
ζ= −ρζ.(8)
é(5) ª¥1˜ª¦÷V••Lie ê,¿|^Ù5Ÿk
L
V
S(X,ζ) =(L
V
S)(X,ζ)+S(L
V
X,ζ)+S(X,L
V
ζ)
= (L
V
S)(X,ζ)+g(L
V
X,Qζ)+S(X,L
V
ζ)
= (L
V
S)(X,ζ)+2ng(L
V
X,ζ)+S(X,L
V
ζ)
= 2n(L
V
η)X+2nη(L
V
X),
DOI:10.12677/pm.2022.12101781652nØêÆ
¸Ú9§4ï¤
l
(L
V
S)(X,ζ)+S(X,L
V
ζ) = 2n(L
V
η)X.
|^(8) ªŒ±
(L
V
S)(X,ζ) = 4nρη(X).
Óž-Y= ζ“\(7)ª¥,k
g(∇
X
Dρ,ζ) = ση(X),(9)
Ù¥σ=
4nρ−∆ρ
1−2n
.,˜•¡3(6)ª¥-Y= Z= ζ
(L
V
R)(X,ζ)ζ=g(∇
X
Dρ,ζ)ζ−g(∇
ζ
Dρ,ζ)X
+η(X)∇
ζ
Dρ+∇
X
Dρ.
(10)
‰(4) ª¦÷V••Lie ê¿ò(8)ª“\
(L
V
R)(X,ζ)ζ= 2ρ(−X−η(X)ζ).(11)
Ïdò(9) ªÚ(11) ª“\(10) ª¥k
∇
X
Dρ= 2ρ(−X−η(X)ζ)−σX.(12)
3(12) ª¥,-X= e
i
,¿éi¦Úk∆ρ= 4nρ,lσ= 0.Ïdk
∇
X
Dρ= 2ρ(−X−η(X)ζ).(13)
,éL
V
η= ρηü>¦‡©d(d´‡©Žf),5¿dÚL
V
Œ†^S,u´k
L
V
dη= dρ∧η.
Ï•dη3M¥´•",¤±k
dρ= (ζρ)η.
‰þª?1‡©,Ï•d
2
= 0,¤±é∀X∈X(M) Œ±
X(ζρ) = −ζ(ζρ)η(X).
dž,|^(3) ª∇
ζ
ζ=0,…3(13) ª¥-X=ζŒ±Ñ∇
ζ
Dρ=0,éζρ=g(ζ,Dρ)
¦÷ζ••Cêkζ(ζρ)= 0,=ζρ´˜‡ ~ê.u´-Dρ=Kζ(Ù¥K=ζρ),é
Ù¦÷X••Cê,Šâ(3)ªk
∇
X
Dρ= K(−X−η(X)ζ).
òþª†(13) ªƒ'Œ±wÑK=2ρ=constant,ρ= constant,?˜Úρ=0,ÏdV
DOI:10.12677/pm.2022.12101781653nØêÆ
¸Ú9§4ï¤
´Killing •þ|,½ny..
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(12161078)§[ ‹ŽpÆM#UåJ,‘8(2019B-045)§[
‹Ž‰EOy]Ï‘8(20JR5RA515)"
ë•©z
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DOI:10.12677/pm.2022.12101781654nØêÆ

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