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

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PureMathematics nØêÆ,2022,12(2),257-263

PublishedOnlineFebruary2022inHans.//www.abtbus.com/journal/pm

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

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Á ‡

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 ÛÜÈiù6 /ŒØCf6 /R†©ÙD

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 ŒÈ¿‡^‡A

W ∈D

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i ù6/§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"

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.1951 c,

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

= I( F6= ±I), =’È(, ¿3ÛÜ‹IXe‰ÑFAŠ1 Ú−1 éAü‡Ö©Ù

Ú T

− 1

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

1964 c,Hsu é’È6/Š•¡ïÄ[4].

1960 c, Tachibana 3ÛÜ‹IXe^’È(F‰ÑÛÜÈiù6/½Â[5].1981 c,

Adati 3ïÄÛÜÈiù6/f6/¯Kžuy, ÛÜ‹IXeÛÜÈiù6/†Ùf6/'

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

ØC f6/[6].

1984 c, Bejancu ½ÂÛÜÈiù6/ŒØCf6/[7], d,Matsumoto[8], Atceken

[9–13] ÚGerdan[13,14] <ïÄ´Lù˜af6/.Bejancu ÏL1Ä/ªB‰ÑÛÜ

È iù6/ŒØCf6/R†©ÙD

⊥

Ú Y²©ÙD©OŒÈ¿‡^‡[7].2003c,

Atceken •|^1Ä/ªB‰ÑR†©ÙD

⊥

Ú Y²©ÙD©OŒÈØÓ¿‡^

‡ [9],2005c,Atceken/Ï•þ©)•{qDÚD

⊥

 ©òÏLWeingartenúª‰ÑR†©ÙD

⊥

 ŒÈ¿‡^‡A

W ∈D

⊥

, ùí2

Atceken 3©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

f 6/,D

⊥

 ÿ/ŒØCf6/,D ÿ/ŒØCf6/[10].

DOI:10.12677/pm.2022.122030258 nØêÆ

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 ©,˜‡¯K´&ÄÛÜ Èiù6/ŒØC f6/¤•·Üÿ /ŒØC f6/¿

Ä u±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

M ÚTM ©OL«(

M, ¯g)Ú(M,g)ƒm,

∇ Ú∇ ©O´(

M, ¯g)Ú(M,g)þéä.-

⊥

M L«(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 .

(

M, ¯g)å‰\f6/(M,g)Weingartenúª•[15]:

∇

ξ =−A

X +∇

⊥

ξ, (2)

Ù ¥X∈TM,ξ∈T

⊥

M ,A

X ∈TM ,∇

⊥

ξ ∈T

⊥

M .

∇

Y 'u(M,g ){•©þB (X,Y )Ú

∇

ξ 'u(M,g )ƒ•©þA

X kXe'X[15]:

¯ g(A

X,Y ) =¯g(B(X,Y),ξ) =¯g(X,A

Y ). (3)

½½½ ÂÂÂ2.1[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/.

½½½ ÂÂÂ2.2[5](

M, ¯g,F)´’Èiù6/,e

∇ F= 0,K¡(

M, ¯g,F)´ÛÜÈiù6/.

½½½ ÂÂÂ2.3[7](M,g)´ÛÜÈiù6/(

M, ¯g,F)åE\f6/,e(M,g)R†©Ù

⊥

Ú Y²©ÙD÷v:

(i) TM= D

⊥

⊕D ;

(ii) F( D) = D;

(iii) F( D

⊥

) ⊂T

⊥

M ,

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.2022.122030259 nØêÆ

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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).

½½½ ÂÂÂ2.4[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/.

½½½ ÂÂÂ2.5[9](M,g)´ÛÜÈiù6/(

M, ¯g,F)ŒØCf6/,e∀X∈D,Z∈D

⊥

, k

B (X,Z ) = 0,K¡(M,g )´(

M, ¯g,F)·Üÿ/ŒØCf6/.

½½½ ÂÂÂ2.6[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

ØC f6/.

½½½ ÂÂÂ2.7[16]D

• m‘ ¢‡ ©6/Ml‘ ©Ù,e∀X,Y∈D

, kLie )Ò

[ X,Y] ∈D

K ¡©ÙD

÷ vFrobenius^‡.

ÚÚÚ nnn2.1[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

∇

( F

Y ) =F (

∇

Y ). (5)

··· KKK2.3[9](M,g)´ÛÜÈiù6/(

M, ¯g,F)ŒØCf6/,K∀Z,W∈D

⊥

, k

W =−A

Z. (6)

ÚÚÚ nnn2.2[9](M,g)´ÛÜÈiù6/(

M, ¯g,F)ŒØCf6/,T

⊥

M =F (D

⊥

) ⊕V, K

⊥

 ŒÈ…=∀X∈D,Z∈D

⊥

, k

B (X,Z )∈V. (7)

3. ŒØCf6/

Œ ØCf6/ƒmÚ©Ùk›©—ƒ'X,ÏdÏL©Ù5•xŒ ØCf6/´š~k¿

Â .!Ì‡ &ÄÛÜÈiù6/ŒØCf6/R†©ÙD

⊥

 ŒÈ¿‡^‡±9ŒØ

C f6/¤•·Üÿ /ŒØC f6/¿©^‡.

··· 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.2022.122030260 nØêÆ

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yyy ²²²Šâ½Â2.3,∀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)

du W,Z∈D

⊥

⊂ TM⊂ T

M ,A^(5),Œ±

∇

( FW) −

∇

( FZ) = F(

∇

W )−F (

∇

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

Š â(6),∀Z,W∈D

⊥

, k

− 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)

Š â(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 ).

½½½ nnn3.1(M,g)´ÛÜÈiù6/(

M, ¯g,F)ŒØCf6/,KD

⊥

 ŒÈ…=

∀ Z,W∈D

⊥

, kA

W ∈D

⊥

yyy ²²²(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.

½½½ nnn3.2(M,g)´ÛÜÈiù6/(

M, ¯g,F)ŒØCf6/,eD

⊥

 ŒÈ,…

F (D

⊥

) = T

⊥

M ,K(M,g )´(

M, ¯g,F)·Üÿ/ŒØCf6/.

yyy ²²²(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.2022.122030261 nØêÆ

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d ·K3.1Ú(17),´

¯ g([Z,W],FX) = 2¯g(B(X,W),FZ).(18)

Š âÚn2.2,D

⊥

 ŒÈduB(X,W)∈V,2dF(D

⊥

)= T

⊥

M ,=T

⊥

M ¥Ø•3©

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

M, ¯g,F)·Üÿ/ŒØCf6/,

½ n y.

ííí ØØØ3.1(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.

ííí ØØØ3.4(M,g)´ÛÜÈiù6/(

M, ¯g,F)Dÿ/ŒØCf6/,K∀X∈D,

ξ ∈T

⊥

M ,k

X ∈D

⊥

4. (Ø

⊥

 ŒÈ¿‡^‡,=

W ∈D

⊥

Û ÜÈiù6/·Üÿ/ŒØCf6/.

Ä 7‘8

I [g,‰ÆÄ7(11761069).

ë •©z

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