"Science and Education" Scientific Journal
March 2021 / Volume 2 Issue 3
YOPIQ 1-FORMALAR HOSIL QILGAN QATLAMALAR
Durdona G’ayrat qizi G’afforova
durdonagaffarovaa@gmail.com
O’zbekiston milliy universiteti
Annotatsiya: Bu maqolada riman qatlamasini urinma vektor maydoni killing
vektor maydoni bo’lishi o’rganilgan. Qatlamalar, riman qatlamasi ta’riflari
keltirilgan. Shuningdek, maqolada silliq ko’pxillik, qatlamalar
maydonida ochib
berilgan.
Kalit so’zlar: riman qatlamasi, Levi-Civita bog’lamasi, vektor maydon, silliq
ko’pxillik, yopiq 1-forma, qatlama, killing vektor maydoni.
FOLIATIONS DEFINED BYCLOSED DIFFERENTIAL 1-FORM
Durdona Gayrat qizi Gafforova
durdonagaffarovaa@gmail.com
National University of Uzbekistan
Abstract: In this article, the attempt to vector the riman layer is studied to be the
killing vector field. Layers, rhymes, descriptions are given. The article also reveals a
smooth polygon
in the area of the folds.
Keywords: Riemann layer, Levi-Civita bond, vector field, smooth polynomial,
closed 1st form, folding, killing vector field.
F = L ;  B
Ta’rif 1.1
ning chiziqli bog’lanishli qism to’plamlari oilasi
berilgan bo’lsin. Agar bu oila quyidagi uchta shartni qanoatlantirsa;
L =M
( F ): B 
;
L L = 
( F ): barcha  ,   B ,    lar uchun 
bo’lsin;
( F ): Har bir p  M
(s)
nuqta uchun shunday (U  , )  A , p U  lokal
koordinatalarni tanlash mumkinki,    B uchun U  L   bo’lsa u holda
 (U  L ) to’plamning chiziqli bog’lanishli komponentalari:
( x , x ,..., x )  (U ) : x
1
2
n
k +1
= ck +1, xk +2 = ck +2 ,..., xn = cn 
ko’rinishga ega bo’ladi, bunda ck +1, ck +2 ,..., cn lar o’zgarmas sonlar,
s
F - k o’lchovli ( 0  k  n ) C - qatlama deyiladi.
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"Science and Education" Scientific Journal
March 2021 / Volume 2 Issue 3
Yuqorida keltirilgan ta’rifning 1-qismida: barcha L qatlamalar M to’plamni
qoplaydi, bu yerda    . 2-qismida: ixtiyoriy  ,   B o’zaro bir-biridan farqli
L
sonlar uchun L va  o’zaro kesishmasligi aytib o’tilgan. Buni quyidagi 1.1-rasmda
yaqqol ko’rishimiz mumkin:
L - to’plam
qatlamaning qatlami deyiladi.
n
M n ko’pxillikda F qatlamning berilishi ( M , F ) simvol bilan aniqlanadi. (1) va
n
(2) shartlar M ko’pxillikning o’zaro kesishmaydigan qatlamlardan tuzilganligini
n
ifodalaydi. (3) shart k-o’lchamli qatlamning yuqorida yozilgan R dagi kabi lokal
ko’rinishga ega ekanligini ifodalaydi.
M = R ( x1, x2 ) \ ( 0,0) F = L : L = ( x1 , x2 ) : x12 + x22 =  
Misol 1.1.
,
. Bu oilani
qatlama hosil qilishini ko’rsataylik. Qatlamaning ta’rifiga ko’ra uchta shartga
tekshiramiz.
L = R 2
( F ): 
;
   2  L1 L 2 = 
( F ): 1
 y1 = x1

2
2
2
U
=
R
,
x

0
F
 y2 = x1 + x2 , x2  0
+
2

( ):
Ta’rifdagi uchta shart bajarildi, demak bu oila qatlama hosil qilar ekan. (1.2rasm)
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"Science and Education" Scientific Journal
March 2021 / Volume 2 Issue 3
Tarif.1.2 Agar biron bir nuqtada -qatlamaning qatlamiga perpendikulyar chiziq
o’tkazilsa bu chiziq barcha qatlamlarga perpendikulyar bo’lsa -qatlama Riman
qatlamasi deyiladi. Riman qatlamasi birinchi bo’lib Reinhart tomonidan kiritilgan va
o’rganilgan.
Teorema 1.1. Agar f : M → N -silliq akslantirishimiz submersiya bo’lsa
L = p  M : f ( p ) = q
(dimM=n, dimN=m). U holda shunday q  N , q 
to’plam-dan
L
k = ( n − m)
olingan nuqta topiladiki, q M ko’pxillikda o’lchovi
ga teng bo’lgan
qatlamni aniqlaydi.
Ta’rif 1.3. Chiziqli fazoda
bo’lsin. Bunda
bazisda
qo’shma basiz
egri chiziqli vektor Differensial 1-forma deyiladi. Har
qanday 1-forma o’zining ko’rinishiga ega
bu yerda
-silliq
funksiya. 1-forma yopiq bo’lishi uchun uning differensiali nolga teng bo’lishi kerak.
Agar berilgan forma biror funksiyaning differensiali bo’lsa, bunday forma aniq forma
deyiladi. Shuning uchun aniq formalar yopiq forma bo’ladi, teskarisi shart emas.
Quyidagi teorema shuni ko’rsatadiki,
kompakt Riman ko’pxilligida yopiq
qatlamlashish bo’ladi.
Teorema 1.2
1-forma izometrik egrilik bilan geodezik
1-forma doimiy egrilikning silliq bog’langan kompakt
Riman ko’pxilligi bo’lsin. Agar
bo’lsa u holda
-Riman qatlamalari silliq bog’langan
Riman qatlamalari yopiq 1-forma bilan berilgan
- qatlamalari o’zaro izometrik to’la geodezik qatlama hosil qiladi.
Misol 1.2.
haqiqiy sonlar bo’lgan
differentsial shaklini ko’rib chiqing. Ushbu shakl
-o’lchovli torus
bo’yicha differentsial shaklni keltirib chiqaradi, bu erda
tenglama
-qatlamaning
ga tenglashtirganligini belgilaydi. Agar
raqamlarning ratsional sonlar to’plami
guruhi
Misol 1.3
www.openscience.uz
- butun sonlar to’plami.
, bu erda
-metrikfunksiya
15
ga teng bo’lsa, u holda
yig’indilar.
"Science and Education" Scientific Journal
March 2021 / Volume 2 Issue 3
Teorema:1.3
-silliq bog’lanishli riman ko’pxilligida
yopiq 1-forma
yordamida hosil qilingan Riman qatlamasi bo’lsin.U holda qatlamalarning urinma
vector maydonlari - Killing vector maydon bo’ladi.
Foydalanilgan adabiyotlar
1. Ghys, E., ClassiÖcation des feuetagestotalementgeodesiques de codimension
un, Comment.Math.Helvetici., 58(1983), 543-572.
2. Gromoll, D., Klingenberg, W. and Meyer, W., Riemannian geometry in the
large. (Russian), Moscow, "Mir", 1971.
3. Hermann, R., A su¢ cient condition that a mapping of Riemannian manifolds
be a Öber bundle, Proc. Amer. Math. Soc., 11(1960), 236-242.
4. Hermann, R., The dierential geometry of foliations, Ann. of Math., 72(1960),
445- 457
5. Narmanov, A. and Kaypnazarova, G., Metric functions on riemannian
manifold,Uzbek math. Journal, 2(2010), 113-121.
6. Narmanov, A. and Kaypnazarova, G., Foliation theory and its applications, J.
PureAppl. Math., 2(1)(2011), 112 - 126.
7. Tondeur, Ph., Foliations on Riemannian manifolds, Springer-Verlag, 1988.
References
1. Ghys, E., ClassiÖcation des feu¬etagestotalementgeodesiques de
codimension un, Comment.Math.Helvetici., 58 (1983), 543-572.
2. Gromoll, D., Klingenberg, W. and Meyer, W., Riemannian geometry in the
large. (Russian), Moscow, "Mir", 1971.
3. Hermann, R., A su ¢ cient condition that a mapping of Riemannian manifolds
be a Öber bundle, Proc. Amer. Math. Soc., 11 (1960), 236-242.
4. Hermann, R., The dierential geometry of foliations, Ann. of Math., 72 (1960),
445- 457
5. Narmanov, A. and Kaypnazarova, G., Metric functions on riemannian
manifold, Uzbek math. Journal, 2 (2010), 113-121.
6. Narmanov, A. and Kaypnazarova, G., Foliation theory and its applications, J.
PureAppl. Math., 2 (1) (2011), 112 - 126.
www.openscience.uz
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"Science and Education" Scientific Journal
March 2021 / Volume 2 Issue 3
7. Tondeur, Ph., Foliations on Riemannian manifolds, Springer-Verlag, 1988.
www.openscience.uz
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