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PureMathematicsnØêÆ,2022,12(12),2163-2169
PublishedOnlineDecember2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1212232
Lorentz˜m¥‡-¡þRicciáf

∗
§§§‡‡‡
†
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2022c1125F¶¹^Fϵ2022c1221F¶uÙFϵ2022c1230F
Á‡
©ïÄLorentz ˜mE
n+1
1
¥‡-¡þ ±§ ˜•þƒ••³•þ|Ricci áf"3
‡-¡/GŽfŒézb½e§‡-¡–õkü‡ØƒÓÌ-Ç"
'…c
Ricciáf§‡-¡§Lorentz˜m§/GŽf§Ì-Ç
RicciSolitonsonHypersurfacesofLorentz
Space
YangYang
∗
,ChaoYang
†
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Nov.25
th
,2022;accepted:Dec.21
st
,2022;published:Dec.30
th
,2022
Abstract
Inthispaper,westudyRiccisolitonsonhypersurfacesofLorentzspaceE
n+1
1
by
∗1˜Šö"
†ÏÕŠö"
©ÙÚ^:,‡.Lorentz˜m¥‡-¡þRicciáf[J].nØêÆ,2022,12(12):2163-2169.
DOI:10.12677/pm.2022.1212232
§‡
takingthepotentialvectorfieldasthetangentcomponentofthepositionvectorof
thehypersurfaces.Undertheassumptionthatthehypersurfaceshavediagonalizable
shapeoperators,weprovethatthehypersurfaceshaveatmosttwodistinctprincipal
curvatures.
Keywords
RicciSolitons,Hypersurfaces,LorentzSpace,ShapeOperator,PrincipalCurvatures
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
(M,g) •–iù6/,e•3Mþ1wƒ•þ|ξ,9λ∈R,¦
1
2
L
ξ
g+Ric = λg,
(1)
Ù¥L
ξ
g´Ýþg÷ξ••Lieê,Ric´MþRicci-ÇÜþ,K¡(M,g,ξ,λ)•Ricci
áf,ξ•áf(M,g,ξ,λ) ³•þ|,λ•áf~ê.λ>0(=0,½<0 )ž,¡Ricci 
áf(M,g,ξ,λ) •Â (-½,½*Ü).AO/,XJL
ξ
g= 0,K¡Ricci áf´²….
Ricci áfAÛ(ÉAÛÆ[2•'5[1–15],Chen[7]ÚDeshmukh[8]ïÄ
m¥‡-¡þ±§ ˜•þƒ•Ü©•³•þ|Ricci áf,y‡-¡–õkü
‡ØÓÌ-Ç,…édaRicciáf?1©a.CÏ,Demirci[11]•Minkowski˜mE
4
1
¥‡-¡þdaRicciáf,3‡-¡/GŽfŒézb½e,Óy‡-¡–õkü
‡ØÓÌ-Ç,g,‡¯.ù‡(Ø3˜„‘êLorentz˜mE
n+1
1
¥´Ä¤á.
©ïÄLorentz ˜mE
n+1
1
¥/GŽfŒéz‡-¡Mþ±§ ˜•þ|ƒ•Ü
©•³•þ|˜aRicciáf,éþã¯K‰Ñ’½£‰, •Ò´,^©z[11]•{,=ò½
Â'uRicci-ÇÜþªfÚTaRicciáf•3¿‡^‡?1OŽé',3˜„
‘êLorentz˜mE
n+1
1
¥,‡-¡/GŽfŒézb½e,ÓŒ±y‡-¡–õk
ü‡ØÓÌ-Ç.ù‡(JØ^•›‘ê,3˜ƒ'OŽ9˜AÛ(¥Ñk-‡¿Â.
©Œ—SüXe:k0ƒ'Ä:•£,ÙgÚ\y²©(JI‡ü‡Ún,•‰Ñ
©̇½n9y².
DOI:10.12677/pm.2022.12122322164nØêÆ
§‡
2.ý•£
Lorentz˜m´•K•I•1–mE
n+1
1
,ÙþDƒ•I•1IO–î¼Ýþ
˜g= −dx
2
0
+dx
2
1
+...+dx
2
n
.
x:(M,g)→(E
n+1
1
,˜g)´˜‡åE\,N´‡-¡M˜‡ü {•þ|,ε=
˜g(N,N) = ±1.^
˜
∇L«E
n+1
1
þLevi-Civitaéä,KéMþ?¿1wƒ•þ|X,k
˜
∇
X
x= X.
(2)
^∇L«MþLevi-Civitaéä,AL«‡-¡M÷N••/GŽf,KéMþ?¿1
wƒ•þ|X,Y,k
˜
∇
X
Y= ∇
X
Y+εg(AX,Y)N,
˜
∇
X
N= −AX.
(3)
dþª
Ric(X,Y) = nHg(AX,Y)−g(AX,AY),
(4)
Ù¥H=
1
n
trA•‡-¡M²þ-Ç.
‡-¡MþCodazzi•§•:é6/Mþ?¿1wƒ•þ|X,Y,k
(∇A)(X,Y) = (∇A)(Y,X),
(5)
Ù¥(∇A)(X,Y) = ∇
X
(AY)−A(∇
X
Y).
X´6/Mþ1wƒ•þ|,Ýþg÷X••Lieê½Â•[16]
(L
X
g)(Y,Z) = X(g(Y,Z))−g([X,Y],Z)−g(Y,[X,Z]),
(6)
Ù¥Y,Z´Mþ?¿1wƒ•þ|.
∇fL«fFÝ,K
g(∇f,X) = df(X) = X(f).
e‡-¡M/GŽfŒéz,
K•3MþÛÜIOIe{e
1
,e
2
,···,e
n
},=
g(e
1
,e
1
) = −ε,g(e
i
,e
i
) = 1,i= 2,···,n,
g(e
i
,e
j
) = 0,i,j= 1,2,···,n,i6= j,
¦Ae
i
= a
i
e
i
,i= 1,2···n,Ù¥a
1
,a
2
,···,a
n
•Mþ1w¼ê.
DOI:10.12677/pm.2022.12122322165nØêÆ
§‡
3.̇(J
y²©̇(JI‡˜O ŽŠ•Á=,Ù¥é-‡˜‡ªf´˜aRicci áf•3
¿‡^‡,…ùaRicci áf´î¼˜m¥‡-¡þ±§ ˜•þƒ•Ü©•³•þ|,
•¤½ny²,·‚I‡±eÚn1ÚÚn2.
ÚÚÚnnn1x: (M,g) →(E
n+1
1
,˜g)´ln‘–iù6/Mn+1 ‘Lorentz˜mE
n+1
1

åE\,Ké6/Mþ?¿1wƒ•þ|X,k
∇
X
x
T
= X+ερAX, ∇ρ= −Ax
T
,
(7)
Ù¥x
T
´xƒ•Ü©,ρ=˜g(x,N),∇ρ´ρFÝ.
y² ˜•þxŒ±©)•
x= x
T
+ερN,
(8)
Ù¥x
T
´xƒ•Ü©,ρ=˜g(x,N) ,ε=˜g(N,N) .
ò(8)ª“\(2)ª,(Ü(3)ª
X=
˜
∇
X
(x
T
+ερN)
=
˜
∇
X
x
T
+ε(X(ρ)N+ρ
˜
∇
X
N)
= ∇
X
x
T
+εg(AX,x
T
)N+εX(ρ)N−ερAX
= ∇
X
x
T
−ερAX+εg(X,Ax
T
)N+εg(∇ρ,X)N,
Ù¥X•Mþ?¿1wƒ•þ|.*þª¥ƒ•Ü©Ú{•Ü©
∇
X
x
T
= X+ερAX,g(∇ρ,X) = −g(X,Ax
T
).
qÏþªé?¿X¤á,¤±∇ρ= −Ax
T
.
ÚÚÚnnn2Lorentz˜mE
n+1
1
¥‡-¡Mþ•3Ricciáf(M,g,x
T
,λ)…=M
Ricci-ÇÜþ÷v
Ric(X,Y) = (λ−1)g(X,Y)−ερg(AX,Y),
(9)
Ù¥X,Y•6/Mþ?¿1wƒ•þ|.
y²ky7‡5.x: (M,g) →(E
n+1
1
,˜g)´åE\,(M,g,x
T
,λ) ´Ricci áf.éu
Mþ?¿1wƒ•þ|X,Y,|^(6) ª•(L
x
T
g)(X,Y),(ÜéäƒN5,ÃL59(7)
DOI:10.12677/pm.2022.12122322166nØêÆ
§‡
ª
(L
x
T
g)(X,Y) = x
T
(g(X,Y))−g([x
T
,X],Y)−g(X,[x
T
,Y])
= g(∇
x
T
X,Y)+g(X,∇
x
T
Y)−g(∇
x
T
X,Y)
+g(∇
X
x
T
,Y)−g(X,∇
x
T
Y)+g(X,∇
Y
x
T
)
= g(∇
x
T
X,Y)+g(X,∇
x
T
Y)−g(∇
x
T
X,Y)
+g(X+ερAX,Y)−g(X,∇
x
T
Y)+g(X,Y+ερAY)
= g(X,Y)+ερg(AX,Y)+g(X,Y)+ερg(X,AY)
= 2g(X,Y)+2ερg(AX,Y).
(10)
,˜•¡,dáf•§•
(L
x
T
g)(X,Y) = 2λg(X,Y)−2Ric(X,Y).

Ric(X,Y) = (λ−1)g(X,Y)−ερg(AX,Y).
2y¿©5.é6/Mþ?¿1wƒ•þ|X,Y,k
Ric(X,Y) = (λ−1)g(X,Y)−ερg(AX,Y).
5¿(10)ª•,¤á,(Üþªk
1
2
(L
x
T
g)(X,Y) = g(X,Y)+ερg(AX,Y)
= −Ric(X,Y)+λg(X,Y),
=
1
2
L
x
T
g+Ric= λg.
l(M,g,x
T
,λ) ´Ricci áf.
½½½nnnx: (M,g) →(E
n+1
1
,˜g)´ln‘–iù6/Mn+1 ‘Lorentz˜mE
n+1
1
å
E\.bMäkŒéz/GŽfA,…(M,g,x
T
,λ) ´Ricci áf,Ù¥x
T
• ˜•þ|
xƒ•Ü©,K‡-¡M–õkü‡ØÓÌ-Ç.
yyy²²²du‡-¡M/GŽfŒéz,•3ÛÜIOIe|
{e
1
,e
2
,···,e
n
},¦Ae
i
=a
i
e
i
,i=1,2,···,n,a
1
,a
2
,···,a
n
•Mþ1w¼ê.OŽ
trA= a
1
+a
2
+···+a
n
,(ÜH=
1
n
trA,kH=
1
n
(a
1
+a
2
+···+a
n
).d(4) ª
DOI:10.12677/pm.2022.12122322167nØêÆ
§‡









Ric(e
1
,e
1
) = −εa
1
Σ
n
k=2
a
k
,
Ric(e
i
,e
i
) = a
i
P
k6=i
a
k
,i= 2,···,n.
(11)
,˜•¡,|^(9)ªOŽ







Ric(e
1
,e
1
) = ε(1−λ)+ρa
1
,
Ric(e
i
,e
i
) = λ−1−ερa
i
,i= 2,···,n.
(12)
é'(11)ª9(12)ª
λ−1−ερa
i
= a
i
Σ
k6=i
a
k
,i= 1,···,n.
dþªŒ•
(a
i
−a
j
)(Σ
k6=i,j
a
k
+ερ) = 0.
e‡-¡M–õäk3 ‡½3‡±þØÓÌ-Ç,Ø”a
1
,a
2
,a
3
p؃,þª¥i= 1,j=
2,3,k
Σ
k6=1,2
a
k
= −ερ,(13)
Ú
Σ
k6=1,3
a
k
= −ερ.(14)
(13) ªÒ†müý©O~(14) ªÒ†müý,a
2
= a
3
,gñ.‡-¡M–õkü
‡ØÓÌ-Ç.
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11761061),[‹Ž‰EOy‘8(20JR5RA515),Ü“‰ŒÆ“
c“‰ïUåJ,‘8(NWNU-LKQN2019-23).
ë•©z
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