On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature
We prove the nonexistence of isometric immersion of geometries $\text{Nil}^3$, $\widetilde{SL}_2$ into the four-dimensional space $M_c^4$ of the constant curvature $c$. We establish that the geometry $\text{Sol}^3$ cannot be immersed into $M_c^4$ if $c \neq -1$ and find the analytic immersion of thi...
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| Дата: | 2005 |
|---|---|
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| Формат: | Стаття |
| Мова: | Російська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2005
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3609 |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509727076646912 |
|---|---|
| author | Masal'tsev, L. A. Масальцев, Л. А. Масальцев, Л. А. |
| author_facet | Masal'tsev, L. A. Масальцев, Л. А. Масальцев, Л. А. |
| author_sort | Masal'tsev, L. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:59:42Z |
| description | We prove the nonexistence of isometric immersion of geometries $\text{Nil}^3$, $\widetilde{SL}_2$ into the four-dimensional space $M_c^4$ of the constant curvature $c$. We establish that the geometry $\text{Sol}^3$ cannot be immersed into $M_c^4$ if $c \neq -1$ and find the analytic immersion of this geometry into the hyperbolic space $H^4(-1)$.
|
| first_indexed | 2026-03-24T02:45:42Z |
| format | Article |
| fulltext |
K�O�R�O�T�K�I���P�O�V�I�D�O�M�L�E�N�N�Q
UDK 514
L. A. Masal\cev (Xar\kov. nac. un-t)
OB YZOMETRYÇESKOM POHRUÛENYY TREXMERNÁX
HEOMETRYJ SL
!!
2, Nil, Sol V ÇETÁREXMERNOE
PROSTRANSTVO POSTOQNNOJ KRYVYZNÁ
We prove the nonexistence of isometric immersion of geometries Nil 3
, SL
!
2 into the four-dimensional
space Mc
4 of the constant curvature c. We establish that the geometry Sol 3 cannot be immersed into
Mc
4 if c ≠ –1 and find the analytic immersion of this geometry into the hyperbolic space H 4 1( )− .
Dovedeno neisnuvannq izometryçnoho zanurennq heometrij Nil 3
, SL
!
2 u çotyryvymirnyj pros-
tir Mc
4
stalo] kryvyny c. Vstanovleno, wo heometriq Sol 3
ne moΩe buty zanurena u Mc
4
pry c ≠ –1, i znajdeno ]] analityçne zanurennq v hiperboliçnyj prostir H 4 1( )− .
V nastoqwej stat\e yssleduetsq vopros ob yzometryçeskom pohruΩenyy
trexmern¥x heometryj Terstona SL
!
2 , Nil, Sol ·1, s. 112‚ v çet¥rexmernoe
prostranstvo Mc
4
postoqnnoj sekcyonnoj kryvyzn¥ c. Analytyçesky zadaça
svodytsq k yssledovanyg razreßymosty system¥ uravnenyj Haussa y Kodaccy v
kaΩdom yz trex sluçaev, hde metryçeskyj tenzor gij y tenzor Rymana qvlqgtsq
yzvestn¥my velyçynamy, a ßest\ neyzvestn¥x komponent bij vtoroho funda-
mental\noho tenzora dolΩn¥ udovletvorqt\ systeme yz ßesty uravnenyj
Haussa y vos\my uravnenyj Kodaccy. Poskol\ku systema uravnenyj qvlqetsq
pereopredelennoj, estestvenno oΩydat\ otrycatel\noho otveta na vopros o
suwestvovanyy lokal\noho yzometryçeskoho pohruΩenyq yzuçaem¥x rymanov¥x
mnohoobrazyj. Tak v dejstvytel\nosty y proysxodyt dlq SL
!
2 y Nil, no v slu-
çae s Sol neoΩydanno okaz¥vaetsq, çto ymeetsq pohruΩenye πtoho prostranst-
va v çet¥rexmernoe hyperbolyçeskoe prostranstvo H4
postoqnnoj otryca-
tel\noj kryvyzn¥ –1. NepohruΩaemost\ Nil v çet¥rexmernoe evklydovo
prostranstvo b¥la dokazana v ·2‚ druhym sposobom. Otmetym, çto pry v¥-
çyslenyy komponent tenzora Rymana Rijkl y symvolov Krystoffelq Γjk
i
m¥
pol\zovalys\ systemoj Maple.
1. NepohruΩaemost\ Nil
3
v Mc
4
. Trexmernaq heometryq Nil
3
predstav-
lqet soboj dejstvytel\nug hruppu Ly s zakonom umnoΩenyq ( , , )( , , )x y z x y z′ ′ ′ =
= ( , , )x x y y z z xy+ ′ + ′ + ′ + ′ s levoynvaryantnoj rymanovoj metrykoj ds2 =
= dx dy dz xdy2 2 2+ + −( ) ·1, s. 123‚.
Teorema 1. Ne suwestvuet yzometryçeskoho klassa C3 pohruΩenyq proyz-
vol\noj oblasty heometryy Nil v prostranstvo postoqnnoj kryvyzn¥ Mc
4 .
Dokazatel\stvo. Systema uravnenyj Haussa pohruΩenyq Nil v Mc
4
ymeet
vyd ·3, s. 182‚
R ijkl = b b b b c g g g gik jl il jk ik jl il jk− + −( ). (1)
© L. A. MASAL|CEV, 2005
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3 421
422 L. A. MASAL|CEV
Podstavlqq v (1) konkretn¥e v¥raΩenyq komponent tenzora Rymana, v¥çyslen-
n¥e v Maple, poluçaem sledugwye ßest\ uravnenyj:
R1212 = x2 3
4
− = b b b c x11 22 12
2 21− + +( ),
R1313 = 1
4
= b b b c11 33 13
2− + ,
R2323 = 1
4
= b b b c22 33 23
2− + ,
(2)
R1213 = – x
4
= b b b b cx11 23 13 12− − ,
R1223 = 0 = b b b b12 23 13 22− ,
R1323 = 0 = b b b b12 33 13 23− .
Esly umnoΩyt\ pqtoe uravnenye system¥ (2) na b33 , a ßestoe uravnenye na b23
y v¥çest\, to poluçym b b b b13 22 33 23
2−( ) = 0. Otsgda s uçetom tret\eho urav-
nenyq, esly c ≠ 1
4
, ymeem b13 = 0. Analohyçno, umnoΩaq pqtoe uravnenye na
b23 , a ßestoe uravnenye na b22 y v¥çytaq, poluçaem b12 = 0. Teper\ moΩno
v¥razyt\ yz pervoho, vtoroho y çetvertoho uravnenyj b22 , b33 , b23 çerez b11 y
podstavyt\ v tret\e uravnenye. V rezul\tate poluçym
b11
2 = c
c
x c
−
− −
0 25
0 752
,
, , b22 = ( , ) ,0 25 0 752
11
1− − −( ) −c x c b ,
b33 = ( , )0 25 11
1− −c b , b23 = − − −( , )0 25 11
1c xb .
Dalee yspol\zuem uravnenye Kodaccy b12 3, = b13 2, , kotoroe, poskol\ku nenu-
lev¥e symvol¥ Krystoffelq est\
Γ22
1 = –x, Γ23
1 = −Γ13
2 = 1
2
, Γ12
2 = −Γ13
3 = x
2
, Γ12
3 = x2 1
2
− ,
pryvodytsq k vydu b xb x b22 23
2
332 1+ + −( ) = 0. Posle podstanovky v poslednee
uravnenye bik poluçaem absurdnoe toΩdestvo. Ostaetsq yssledovat\ otdel\no
sluçaj c = 1
4
. Tohda systema (2) prynymaet vyd
–1 = b b b11 22 12
2− , 0 = b b b11 33 13
2− , 0 = b b b22 33 23
2− ,
(3)
0 = b b b b11 23 13 12− , 0 = b b b b12 23 13 22− , 0 = b b b b12 33 13 23− .
Esly umnoΩyt\ çetvertoe uravnenye system¥ (3) na b12 , a pqtoe uravnenye
na b11 y v¥çest\, to poluçym b b b b13 12
2
11 22−( ) = 0, otkuda s uçetom pervoho
uravnenyq naxodym b13 = 0. UmnoΩaq çetvertoe uravnenye na b22 , pqtoe
uravnenye na b12 y v¥çytaq, poluçaem b23 = 0. Dalee lehko naxodym b33 = 0.
Yspol\zuq uravnenye Kodaccy C1) b32 1, = b31 2, , ymeem b22 = − b11. Rassmotrym
teper\ sledugwye uravnenyq Kodaccy:
C2) b11 2, = b12 1, , yly
∂
∂
b
y
11 =
∂
∂
+b
x
x b12
122
,
C3) b22 1, = b21 2, , yly − ∂
∂
b
y
12 =
∂
∂
+b
x
x b11
112
.
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
OB YZOMETRYÇESKOM POHRUÛENYY TREXMERNÁX HEOMETRYJ … 423
Zapyßem yx v sledugwem vyde:
C2) ∂
∂ ( )y
e bx2 4
11
/ = ∂
∂ ( )x
e bx2 4
12
/ ,
C3) ∂
∂ ( )x
e bx2 4
11
/ = − ∂
∂ ( )y
e bx2 4
12
/ .
Sledovatel\no, funkcyy U = e bx2 4
12
/
y V = e bx2 4
11
/
qvlqgtsq dejstvy-
tel\noj y mnymoj çastqmy nekotoroj analytyçeskoj funkcyy U + i V pere-
mennoho x + i y. No tohda uravnenye Haussa 1 = b b11
2
12
2+ protyvoreçyt tomu, çto
funkcyq ln U V2 2+( ) dolΩna b¥t\ harmonyçeskoj, çto y zaverßaet dokaza-
tel\stvo.
2. NepohruΩaemost\ SL
!
2 v Mc
4
. Trexmernaq heometryq SL
!
2 predstav-
lqet soboj unyversal\nug nakr¥vagwug hrupp¥ SL2 s levoynvaryantnoj
metrykoj. V ·1, s. 114‚ ukazano, çto SL
!
2 moΩno predstavyt\ kak rassloenye
edynyçn¥x vektorov UH
2
v kasatel\nom rassloenyy nad hyperbolyçeskoj
ploskost\g H
2
. Pust\ H
2 : ( x, y), y > 0 s metrykoj ds
H2
2 =
dx dy
y
2 2
2
+
. Tohda
UH
2
s koordynatamy x, y, t qvlqetsq podmnohoobrazyem v kasatel\nom
rassloenyy nad H
2
s metrykoj ds
UH2
2 =
ds D D
H H2 2
2 + v v, , hde v =
= y t y tcos , sin( ) y D iv = = d dxi
jk
i j kv v+ Γ , i = 1, 2 (komponent¥ absolgtnoho
dyfferencyala vektora v v metryke H
2
). S uçetom toho, çto Γ12
1 = Γ22
2 = − 1
y
,
Γ11
2 = 1
y
, poluçaem
ds
UH2
2 =
dx dy dx ydt
y
2 2 2
2
+ + +( )
. (4)
Teorema 2. Ne suwestvuet yzometryçeskoho klassa C
3 pohruΩenyq proyz-
vol\noj oblasty heometryy SL
!
2 v prostranstvo postoqnnoj kryvyzn¥ Mc
4 .
Dokazatel\stvo. Zapyßem systemu uravnenyj Haussa pohruΩenyq metryky
(4) v Mc
4
:
R1212 = − 3
2 4y
= b b b c
y
11 22 12
2
4
2− + ,
R1313 = 1
4 2y
= b b b c
y
11 33 13
2
2− + ,
R2323 = 1
4 2y
= b b b c
y
22 33 23
2
2− + ,
(5)
R1223 = − 1
4 3y
= b b b b c
y
12 23 13 22 3− − ,
R1332 = 0 = b b b b13 32 12 33− ,
R2113 = 0 = b b b b21 13 23 11− .
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
424 L. A. MASAL|CEV
Rassmotrym snaçala sluçaj c ≠ 1
4
. Kak y v sluçae system¥ (2), poluçym b12 =
= b32 = 0. Zatem moΩno najty ostal\n¥e neyzvestn¥e:
b11 = − −1 5 2
4
22
, c
y b
, b33 = 0 25
2
22
, − c
y b
, b13 = 0 25
3
22
, − c
y b
, b22 =
− −1 75
2
, c
y
.
Teper\, yspol\zuq uravnenye Kodaccy b12 3, = b13 2, y uçyt¥vaq sledugwye
znaçenyq symvolov Krystoffelq metryky (4):
Γ12
1 = − 3
2y
, Γ23
1 = – Γ13
2 = − 1
2
, Γ22
2 = – 1
y
,
Γ11
2 = 2
y
, Γ12
3 = 1
2y
, Γ23
3 = 1
2y
,
poluçaem absurdnoe toΩdestvo.
Yssleduem sluçaj c = 1
4
. Systema uravnenyj Haussa (5) prynymaet vyd
b b b11 22 12
2− = − 2
4y
, b b b11 33 13
2− = 0,
b b b22 33 23
2− = 0, b b b b12 23 13 22− = 0,
b b b b13 32 12 33− = 0, b b b b21 13 23 11− = 0.
UmnoΩaq çetvertoe uravnenye poslednej system¥ na b11, a ßestoe urav-
nenye na b12 y sklad¥vaq, s uçetom pervoho uravnenyq poluçaem b13 = 0. Um-
noΩaq çetvertoe uravnenye na b12 , ßestoe uravnenye na b22 y sklad¥vaq,
ymeem b23 = 0. Zatem yz pervoho, vtoroho, tret\eho y pqtoho uravnenyj sleduet
b33 = 0, y ostaetsq odno uravnenye Haussa b b b11 22 12
2− = − 2
4y
y try neyzvestn¥e
funkcyy b11, b12 , b22 peremenn¥x x, y, z , kotor¥e dolΩn¥ ewe udovletvo-
rqt\ ßesty uravnenyqm Kodaccy sledugweho vyda:
C1) b12 3, = b13 2, , yly
∂
∂
−b
z
b12
22
1
2
= 0,
C2) b21 3, = b23 1, , yly
∂
∂
+b
z
b12
11
1
2
= 0,
C3) b11 2, = b12 1, , yly
∂
∂
+b
y
b
y
11 113
2
=
∂
∂
−b
x
b
y
12 222
,
C4) b11 3, = b13 1, , yly
∂
∂
−b
z
b11
21
1
2
= 0,
C5) b22 1, = b21 2, , yly
∂
∂
+b
x y
b22
12
1
2
=
∂
∂
b
y
12 ,
C6) b22 3, = b23 2, , yly
∂
∂
+b
z
b22
21
1
2
= 0.
Uravnenyq C7) b33 1, = b13 3, y C 8) b33 2, = b32 3, udovletvoren¥ toΩdest-
venno. Sravnyvaq uravnenyq C1) y C2), naxodym b22 = – b11, y tohda uravne-
nyq C4) y C6) ravnosyl\n¥. Zatem uravnenyq C3) y C5) moΩno perepysat\ v
sledugwem vyde:
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
OB YZOMETRYÇESKOM POHRUÛENYY TREXMERNÁX HEOMETRYJ … 425
C3) ∂
∂
y
b
y
11 = ∂
∂
x
b
y
12 , C5) ∂
∂
y
b
y
12 = − ∂
∂
x
b
y
11 .
Otsgda vydno, çto funkcyy U =
b
y
12
y V =
b
y
11
dolΩn¥ predstavlqt\ soboj
dejstvytel\nug y mnymug çasty nekotoroj analytyçeskoj funkcyy peremen-
noho x + i y, pryçem yz uravnenyq Haussa sleduet U V2 2+ = 2
5y
. Odnako tohda
funkcyq ln U V2 2+( ) ne budet harmonyçeskoj, çto y zaverßaet dokazatel\stvo
teorem¥ 2.
3. Yzometryçeskoe pohruΩenye Sol3 v Mc
4
. Trexmernaq heometryq Sol
predstavlqet soboj hruppu Ly s zakonom umnoΩenyq
( , , )( , , )x y z x y z′ ′ ′ = ( , , )x e x y e y z zz z+ ′ + ′ + ′−
y s levoynvaryantnoj metrykoj ds2 = e dxz2 2 + e dyz−2 2 + dz2
·1, s. 127‚.
Teorema 3. 1. Ne suwestvuet yzometryçeskoho klassa C
3 pohruΩenyq
proyzvol\noj oblasty heometryy Sol3 v prostranstvo postoqnnoj kryvyzn¥
Mc
4 pry c ≠ –1.
2. Suwestvuet yzometryçeskoe analytyçeskoe pohruΩenye heometryy Sol3
v hyperbolyçeskoe prostranstvo H4 1( )− , naprymer
x z z
x e x
x e x
x e y
x e y
z z
z z
z z
z z
0 0
1
2
3
4
2
1
2
2
1
2
2
1
2
2
1
2
2
0
0
0
0
= +
=
=
=
=
+
+
− −
− −
cosh( )
cos
sin
cos
sin
( )
( )
( )
( )
,
(6)
hde
H4 1( )− = x x x x x x x x x x x0 1 2 3 4 0
2
1
2
2
2
3
2
4
2
01 0, , , , ,( ) − + + + + = − >( )
— verxnqq pola hyperboloyda v psevdoevklydovom prostranstve R4 1, s metry-
koj ds2 = − + + + +dx dx dx dx dx0
2
1
2
2
2
3
2
4
2 , z0 ∈ R — proyzvol\naq postoqnnaq.
Dokazatel\stvo. Systema uravnenyj Haussa pohruΩenyq Sol3
v Mc
4
yme-
et vyd
R1212 = 1 = b b b c11 22 12
2− + ,
R1313 = −e z2 = b b b ce z
11 33 13
2 2− + ,
R2323 = − −e z2 = b b b ce z
22 33 23
2 2− − − ,
(7)
R1223 = 0 = b b b b12 23 13 22− ,
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
426 L. A. MASAL|CEV
R1332 = 0 = b b b b13 32 12 33− ,
R2113 = 0 = b b b b21 13 23 11− .
Tak Ωe, kak y pry reßenyy system¥ (2), lehko poluçyt\, çto pry c ≠ –1
b12 = b13 = b23 = 0. Zatem naxodym ostal\n¥e koπffycyent¥ vtoroj funda-
mental\noj form¥:
b11 = 1 2− c e z , b22 = 1 2− −c e z , b33 =
1
1
+
−
c
c
.
S pomow\g uravnenyq Kodaccy b11 3, = b13 1, lehko pryjty k protyvoreçyg y,
sledovatel\no, pry c ≠ –1 ne suwestvuet yzometryçeskoho pohruΩenyq Sol3
v
Mc
4
. V sluçae, kohda c = –1, systema (7) prynymaet vyd
b b b11 22 12
2− = 2, b b b11 33 13
2− = 0,
b b b22 33 23
2− = 0, b b b b12 23 13 22− = 0,
b b b b13 32 12 33− = 0, b b b b21 13 23 11− = 0.
Yz çetvertoho y ßestoho uravnenyj poslednej system¥ s uçetom pervoho
uravnenyq sleduet b13 = b23 = 0. Zatem yz pervoho, vtoroho, tret\eho y pqtoho
uravnenyj naxodym b33 = 0. Rassmatryvaq systemu uravnenyj Kodaccy, ubeΩ-
daemsq, çto b12 = 0, b11 = c ez
1 , b22 = 2
1c
e z−
(hde c1 — proyzvol\naq postoqn-
naq) udovletvorqgt vsem ym. Tohda, sohlasno yzvestnoj teoreme Bonne,
suwestvuet yzometryçeskoe pohruΩenye Sol3 v H4 1( )− . Posle πtoho netrudno
najty yzometryçeskoe pohruΩenye (6), pryvedennoe v formulyrovke teorem¥.
V zaklgçenye otmetym, çto v obzore A. A. Borysenko ·4‚ dostatoçno polno
osvewen¥ vopros¥ yzometryçeskoho pohruΩenyq prostranstvenn¥x form v
prostranstva postoqnnoj kryvyzn¥, no poka malo yssledovan¥ zadaçy pohruΩe-
nyq trex- y çet¥rexmern¥x heometryçeskyx struktur v sm¥sle Terstona v
prostranstva postoqnnoj kryvyzn¥.
1. Skott P. Heometryy na trexmern¥x mnohoobrazyqx. – M.: Myr, 1986. – 163 s.
2. Rivertz H. J. An obstruction to isometric immersion of the threedimensional Heisenberg group into
R
4 // Prepr. Ser. Pure Math. – 1999. – # 22.
3. ∏jzenxart L. P. Rymanova heometryq. – M.: Yzd-vo ynostr. lyt., 1948. – 316 s.
4. Borysenko A. A. Yzometryçeskye pohruΩenyq prostranstvenn¥x form v rymanov¥ y psev-
dorymanov¥ prostranstva postoqnnoj kryvyzn¥ // Uspexy mat. nauk. – 2001. – 56, v¥p. 3. –
S.S3S–S78.
Poluçeno 01.12. 2003
ISSN 0041-6053. Ukr. mat. Ωurn., 2005, t. 57, # 3
|
| id | umjimathkievua-article-3609 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T02:45:42Z |
| publishDate | 2005 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/dd/b468a2a3c2184f06448b61353674a5dd.pdf |
| spelling | umjimathkievua-article-36092020-03-18T19:59:42Z On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature Об изометрическом погружении трехмерных геометрий $SL_2$, $Nil$, $Sol$ в четырехмерное пространство постоянной кривизны Masal'tsev, L. A. Масальцев, Л. А. Масальцев, Л. А. We prove the nonexistence of isometric immersion of geometries $\text{Nil}^3$, $\widetilde{SL}_2$ into the four-dimensional space $M_c^4$ of the constant curvature $c$. We establish that the geometry $\text{Sol}^3$ cannot be immersed into $M_c^4$ if $c \neq -1$ and find the analytic immersion of this geometry into the hyperbolic space $H^4(-1)$. Доведено неіснування ізометричного занурення геометрій $\text{Nil}^3$, $\widetilde{SL}_2$ у чотиривимірний простір $M_c^4$ сталої кривини $c$. Встановлено, що геометрія $\text{Sol}^3$ не може бути занурена у $M_c^4$ при $c \neq -1$, і знайдено її аналітичне занурення в гіперболічний простір $H^4(-1)$. Institute of Mathematics, NAS of Ukraine 2005-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3609 Ukrains’kyi Matematychnyi Zhurnal; Vol. 57 No. 3 (2005); 421–426 Український математичний журнал; Том 57 № 3 (2005); 421–426 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3609/3949 https://umj.imath.kiev.ua/index.php/umj/article/view/3609/3950 Copyright (c) 2005 Masal'tsev L. A. |
| spellingShingle | Masal'tsev, L. A. Масальцев, Л. А. Масальцев, Л. А. On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature |
| title | On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature |
| title_alt | Об изометрическом погружении трехмерных геометрий $SL_2$, $Nil$, $Sol$ в четырехмерное пространство постоянной кривизны |
| title_full | On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature |
| title_fullStr | On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature |
| title_full_unstemmed | On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature |
| title_short | On Isometric Immersion of Three-Dimensional Geometries $SL_2$, $Nil$ and $Sol$ into a Four-Dimensional Space of Constant Curvature |
| title_sort | on isometric immersion of three-dimensional geometries $sl_2$, $nil$ and $sol$ into a four-dimensional space of constant curvature |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3609 |
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