局部积黎曼流形的半不变子流形 Semi-Invariant Submanifolds of a Locally Product Riemannian Manifold

doi:10.12677/PM.2022.122030

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PureMathematicsnØêÆ,2022,12(2),257-263

PublishedOnlineFebruary2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.122030

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∗

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#õ“‰ŒÆêÆ‰ÆÆ§#õ¿°7à

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Á‡

©Ì‡ïÄÛÜÈiù6/ŒØCf6/ƒ'5Ÿ"Äk§ÏLWeingartenúª§‰Ñ

ÛÜÈiù6/Œ ØCf6/R†©ÙD

⊥

ŒÈ¿‡^‡A

W∈D

⊥

"Ùg§

ÛÜÈiù6/ŒØCf6/¤•·Üÿ/ŒØCf6/¿©^‡"

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iù6/§ÛÜÈ§ŒØCf6/§R†©Ù§·Üÿ/

Semi-InvariantSubmanifoldsofaLocally

ProductRiemannianManifold

LizeBian,ShuwenLi,YongHe

∗

,ChangTian

SchoolofMathematicsScience,XinjiangNormalUniversity,UrumqiXinjiang

Received:Jan.3

,2022;accepted:Feb.3

,2022;published:Feb.10

,2022

Abstract

Inthispaper,westudythesemi-invariantsubmanifoldsofalocallyproductRie-

mannianmanifold.Firstly,forthesemi-invariantsubmanifoldsofalocallyproduct

∗ÏÕŠö"

©ÙÚ^:>áL,oÔ>,Û],X„.ÛÜÈiù6/ŒØCf6/[J].nØêÆ,2022,12(2):257-263.

DOI:10.12677/pm.2022.122030

>áL

Riemannianmanifold,thenecessaryandsuﬃcientconditionsforitsverticaldistribu-

tiontobecompletelyintegrablearegivenbytheWeingartenformula.Secondly,a

suﬃcient conditionisobtained forthesemi-invariantsubmanifolds ofa locally product

Riemannianmanifoldtobemixed-geodesicsubmanifolds.

Keywords

RiemannianManifold,LocallyProduct,Semi-InvariantSubmanifolds,Vertical

Distribution,Mixed-Geodesic

2022byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

3‡©AÛ¥,äkE(J

=−I(I´ðÓN)6/kXš~´L AÛ5Ÿ.1951c,

NijenhuisïÄE(JŒÈ5[1].1958c,Yano&Ä˜‡aqE(1w(

= I(F6= ±I),=’È(,¿3ÛÜ‹IXe‰ÑFAŠ1Ú−1éAü‡Ö©Ù

ÚT

−1

©OŒÈ¿‡^‡[2].

äk’È(‡©6/¡•’È6/.1959c,YanoïÄ’È6/ŒÈ©Ù[3].

1964c,Hsué’È6/Š•¡ïÄ[4].

1960c, Tachibana3ÛÜ‹IXe^’È(F‰ÑÛÜÈiù6/½Â[5].1981c,

Adati3ïÄÛÜÈiù6/f6/¯K žuy,ÛÜ‹IXeÛÜÈiù6/†Ùf6/'

XØéÐ¥y,u´l’È(FN5Ñu½ÂÛÜÈiù6/ØCf6/Ú‡

ØCf6/[6].

1984c, Bejancu½ÂÛÜÈiù6/ŒØCf6/[7],d,Matsumoto[8], Atceken

[9–13]ÚGerdan[13,14]<ïÄ´Lù˜af6/.BejancuÏL1Ä/ªB‰ÑÛÜ

Èiù6/ŒØCf6/R†©ÙD

⊥

ÚY²©ÙD©OŒÈ¿‡^‡[7].2003c,

Atceken•|^1Ä/ªB‰ÑR†©ÙD

⊥

ÚY²©ÙD©OŒÈØÓ¿‡^

‡[9],2005c,Atceken/Ï•þ©)•{qDÚD

⊥

©OŒÈ#¿‡^‡[10].

©òÏLWeingartenúª‰ÑR †©ÙD

⊥

ŒÈ¿‡^‡A

W∈D

⊥

,ùí2

Atceken3©z[10]¥¤A

W= 0ù˜(Ø,4Œ/´LR†©ÙŒÈ5.

ÿ/f6/´iùAÛ¥˜aš~ kïÄdŠf6/,ÛÜÈiù6/ÿ/ŒØCf

6/•´›©-‡.ÿ/ŒØCf6/Ï•Ù ©ÙÀØÓ,kn«a.,=·Üÿ/ŒØC

f6/,D

⊥

ÿ/ŒØCf6/,Dÿ/ŒØCf6/[10].

DOI:10.12677/pm.2022.122030258nØêÆ

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©,˜‡¯K´&ÄÛÜÈiù6/ŒØCf6/¤•·Üÿ/ŒØCf 6/¿

©^‡.

Äu±þ©Û,©ïÄŒ±˜«EäkR†©ÙŒÈŒØCf6/k•

{,„Œ±˜«EÛÜÈiù6/·Üÿ/ŒØCf6/kå».

2.ý•£

!Ì‡0ïÄ¤IÄVg,¿½ÎÒ.

(

M,¯g)Ú(M,g)©O´m+n‘Úm‘iù6/, (M,g)´(

M,¯g)åE\f6/.

T

MÚTM©OL«(

M,¯g)Ú(M,g)ƒm,

∇Ú∇©O´(

M,¯g)Ú(M,g)þéä.-

⊥

ML«(M,g){m,=T

M= TM⊕T

⊥

M,∇

⊥

L«(M,g)þ{éä.

(

M,¯g)å‰\f6/(M,g)Gaussúª•[15]:

∇

Y= ∇

Y+B(X,Y),(1)

Ù¥X,Y∈TM,∇

Y∈TM,B(X,Y) ∈T

⊥

(

M,¯g)å‰\f6/(M,g)Weingartenúª•[15]:

∇

ξ= −A

X+∇

⊥

ξ,(2)

Ù¥X∈TM,ξ∈T

⊥

M,A

X∈TM,∇

⊥

ξ∈T

⊥

∇

Y'u(M,g) {•©þB(X,Y)Ú

∇

ξ'u(M,g)ƒ•©þA

XkXe'X[15]:

¯g(A

X,Y) =¯g(B(X,Y),ξ) =¯g(X,A

Y).(3)

½½½ÂÂÂ2.1[5](

M,¯g)´iù6/,F´(

M,¯g)þ’È(,e∀

Y∈T

M,k

¯g(F

X,F

Y) =¯g(

Y),(4)

K¡(

M,¯g,F)´’Èiù6/.

½½½ÂÂÂ2.2[5](

M,¯g,F)´’Èiù6/,e

∇F= 0,K¡(

M,¯g,F)´ÛÜÈiù6/.

½½½ÂÂÂ2.3[7](M,g)´ÛÜÈiù6/(

M,¯g,F)åE\f6/,e(M,g)R†©Ù

⊥

ÚY²©ÙD÷v:

(i)TM= D

⊥

⊕D;

(ii)F(D) = D;

(iii)F(D

⊥

) ⊂T

⊥

K¡(M,g)´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/.

dim(D

⊥

) = 0 ž, ¡(M,g) •ØCf6/[6]; dim(D) = 0 ž, ¡(M,g) •‡ØCf6/[6].

(M,g)´ŒØCf6/ž,V´F(D

⊥

)3T

⊥

M¥Ö©Ù,=T

⊥

M= F(D

⊥

)⊕V.

···KKK2.1[15]NB:TM×TM→T

⊥

M´(M,g)3(

M,¯g)¥1Ä/ª,KB

DOI:10.12677/pm.2022.122030259nØêÆ

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´é¡…C

∞

V‚5,=∀X,Y∈TM,k:

(i)B(X,Y) = B(Y,X),

(ii)B(aX,bY) = abB(X,Y).

½½½ÂÂÂ2.4[15](M,g)´(

M,¯g)åE\f6/,e(M,g)3(

M,¯g)¥1Ä/ª

B= 0,K¡(M,g)•(

M,¯g)ÿ/f6/.

½½½ÂÂÂ2.5[9](M,g) ´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,e∀X∈D,Z∈D

⊥

B(X,Z) = 0,K¡(M,g)´(

M,¯g,F)·Üÿ/ŒØCf6/.

½½½ÂÂÂ2.6[10](M,g)´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,e∀X,Y∈D,

Z,W∈D

⊥

,kB(Z,W)=0(½B(X,Y) =0),K¡(M,g)´(

M,¯g,F)D

⊥

(½D)ÿ/Œ

ØCf6/.

½½½ÂÂÂ2.7[16]D

•m‘¢‡©6/Ml‘©Ù,e∀X,Y∈D

,kLie)Ò

[X,Y] ∈D

K¡©ÙD

÷vFrobenius^‡.

ÚÚÚnnn2.1[16]D

•m‘¢‡©6/Ml‘©Ù,K©ÙD

ŒÈdu©ÙD

÷vFrobenius^‡.

···KKK2.2[5](

M,¯g,F)´ÛÜÈiù6/,K∀

Y∈T

M,k

∇

Y) = F(

∇

Y).(5)

···KKK2.3[9](M,g)´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,K∀Z,W∈D

⊥

W= −A

Z.(6)

ÚÚÚnnn2.2[9](M,g)´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,T

⊥

M= F(D

⊥

)⊕V,K

⊥

ŒÈ…=∀X∈D,Z∈D

⊥

B(X,Z) ∈V.(7)

3.ŒØCf6/

ŒØCf6/ƒmÚ©Ùk›©—ƒ'X,ÏdÏL©Ù5•xŒØCf6/´š~k¿

Â.!Ì‡&ÄÛÜÈiù6/ŒØCf6/R†©ÙD

⊥

ŒÈ¿‡^‡±9ŒØ

Cf6/¤•·Üÿ/ŒØCf6/¿©^‡.

···KKK3.1(M,g) ´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,K∀Z,W∈D

⊥

,X∈D,k

¯g([Z,W],FX) = 2¯g(A

W,X).(8)

DOI:10.12677/pm.2022.122030260nØêÆ

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yyy²²²Šâ½Â2.3,∀Z,W∈D

⊥

,kFZ,FW∈T

⊥

M,2d(2),Œ

∇

(FW) = −A

Z+∇

⊥

FW,(9)

∇

(FZ) = −A

W+∇

⊥

FZ.(10)

d(9)~(10),Œ

∇

(FW)−

∇

(FZ) = −A

Z+A

W+∇

⊥

FW−∇

⊥

FZ.(11)

duW,Z∈D

⊥

⊂TM⊂T

M,A^(5),Œ±

∇

(FW)−

∇

(FZ) = F(

∇

W)−F(

∇

Z) = F[Z,W].(12)

Šâ(6),∀Z,W∈D

⊥

−A

Z+A

W= 2A

W.(13)

ò(12)Ú(13)“\(11),

F[Z,W] = 2A

W+∇

⊥

FW−∇

⊥

FZ.(14)

Ï•∇

⊥

FW−∇

⊥

FZ∈T

⊥

M,X∈D⊂TM,¤±¯g(∇

⊥

FW−∇

⊥

FZ,X) =0,d(14)Œ±



¯g(F[Z,W],X) = 2¯g(A

W,X).(15)

Šâ(4),¿5¿F

= I,k

¯g(F[Z,W],X) =¯g(F

[Z,W],FX) =¯g([Z,W],FX).(16)

d(16)Ú(15),´

¯g([Z,W],FX) = 2¯g(A

W,X).

½½½nnn3.1(M,g)´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,KD

⊥

ŒÈ…=

∀Z,W∈D

⊥

,kA

W∈D

⊥

yyy²²²(M,g)´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,dÚn2.1Œ•, D

⊥

ŒÈ

du[Z,W] ∈D

⊥

,l∀X∈D,k¯g([Z,W],FX) = 0,2d·K3.1,ùduA

W∈D

⊥

ŒÈ…=A

W∈D

⊥

,½ny.

½½½nnn3.2(M,g)´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,eD

⊥

ŒÈ,…

F(D

⊥

) = T

⊥

M,K(M,g)´(

M,¯g,F)·Üÿ/ŒØCf6/.

yyy²²²(M,g)´ÛÜÈiù6/(

M,¯g,F)ŒØCf6/,Šâ(3), ∀Z,W∈D

⊥

,X∈D,

¯g(A

W,X) =¯g(B(X,W),FZ).(17)

DOI:10.12677/pm.2022.122030261nØêÆ

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d·K3.1Ú(17),´

¯g([Z,W],FX) = 2¯g(B(X,W),FZ).(18)

ŠâÚn2.2,D

⊥

ŒÈduB(X,W)∈V,2dF(D

⊥

)=T

⊥

M,=T

⊥

M¥Ø•3©

ÙV,(18)¥B(X,W) = 0.ld½Â2.5Œ•,(M,g)´(

M,¯g,F)·Üÿ/ŒØCf6/,

½ny.

íííØØØ3.1(M,g)´ÛÜÈiù6/(

M,¯g,F)·Üÿ/ŒØCf6/,K(M,g)R

†©ÙD

⊥

ŒÈ.

ííí ØØØ3.2(M,g)´ÛÜÈiù6/(

M,¯g,F)·Üÿ/ŒØCf6/,K∀Z∈D

⊥

X∈D,ξ∈T

⊥

M,k

Z∈D

⊥

X∈D.

ííí ØØØ3.3(M,g)´ÛÜÈiù6/(

M,¯g,F)D

⊥

ÿ/ŒØCf6/,K∀Z∈D

⊥

ξ∈T

⊥

M,k

Z∈D.

íííØØØ3.4(M,g)´ÛÜÈiù6/(

M,¯g,F)Dÿ/ŒØCf6/,K∀X∈D,

ξ∈T

⊥

M,k

X∈D

⊥

4.(Ø

⊥

ŒÈ¿‡^‡,=

W∈D

⊥

ÛÜÈiù6/·Üÿ/ŒØCf6/.

Ä7‘8

I[g,‰ÆÄ7(11761069).

ë•©z

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