New equations of infinitesimal deformations of surfaces in $E_3$
We establish a condition for two symmetric tensor fields that is necessary and sufficient for the existence of a displacement vector in the case of infinitesimal deformation of a surface in the Euclidean space E 3.
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| Date: | 2010 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508841605595136 |
|---|---|
| author | Potapenko, I. V. Потапенко, І. В. |
| author_facet | Potapenko, I. V. Потапенко, І. В. |
| author_sort | Potapenko, I. V. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2020-03-18T19:39:03Z |
| description | We establish a condition for two symmetric tensor fields that is necessary and sufficient for the existence of a displacement vector in the case of infinitesimal deformation of a surface in the Euclidean space E 3. |
| first_indexed | 2026-03-24T02:31:37Z |
| format | Article |
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UDK 514.752.43
I. V. Potapenko (In-t matematyky, ekonomiky i mexaniky, Odes. nac. un-t)
NOVI RIVNQNNQ INFINITEZYMAL|NYX
DEFORMACIJ POVERXON| V E3
We prove a necessary and sufficient condition for the existence of a displacement vector in the case of
infinitesimal deformation of a surface in the Euclidean space E3 , which must be satisfied by two
symmetric tensor fields.
Dokazano neobxodymoe y dostatoçnoe uslovye, kotoromu dolΩn¥ udovletvorqt\ dva symmetry-
çeskyx tenzorn¥x polq dlq suwestvovanyq vektora smewenyq pry ynfynytezymal\noj defor-
macyy poverxnosty v evklydovom prostranstve E3 .
U klasyçnij teori] poverxon\ u tryvymirnomu evklidovomu prostori vidomog [
teorema Bone [1], v qkij stverdΩu[t\sq, wo qkwo koefici[nty dvox form, odna
z qkyx dodatno oznaçena, zadovol\nqgt\ osnovni rivnqnnq (rivnqnnq Haussa ta
rivnqnnq Majnardi – Petersona – Kodacci), to isnu[ z toçnistg do ruxu ta dzer-
kal\noho vidobraΩennq poverxnq, dlq qko] zaznaçeni formy budut\ koefici[n-
tamy perßo] ta druho] kvadratyçno] formy.
U danij roboti otrymano novu formu umov intehrovnosti dlq isnuvannq vek-
tora zmiwennq infinitezymal\no] deformaci] poverxni v E3 . Cq forma skla-
da[t\sq z rivnqn\, analohiçnyx rivnqnnqm Haussa ta rivnqnnqm Majnardi – Pe-
tersona – Kodacci u klasyçnij teori] poverxon\.
Rozhlqnemo u tryvymirnomu evklidovomu prostori E3 rehulqrnu klasu C k
,
k ≥ 3, poverxng S, homeomorfnu ploskij dvovymirnij odnozv’qznij oblasti z
vektorno-parametryçnym rivnqnnqm
r r x x= ( , )1 2
, (1)
ta ]] infinitezymal\nu rehulqrnu klasu C k
, k ≥ 3, deformacig St :
r r x x t y x xt = +( , ) ( , )1 2 1 2
, (2)
de y x x( , )1 2
— vektor zmiwennq, qkyj [ rehulqrnog klasu C k
, k ≥ 3, vektor-
nog funkci[g v danij oblasti, t — malyj parametr. Çastynni poxidni vektora
zmiwennq zapyßemo u vyhlqdi
y x x P r P ni i i( , ) .
1 2 = + ∗α
α . (3)
Teorema 1. Qkwo rehulqrna poverxnq S klasu C k
, k ≥ 3, v evklidovomu
prostori E3 zazna[ infinitezymal\no] deformaci] (2), to isnugt\ dva sy-
metryçni tenzorni polq αij ta βij , qki zadovol\nqgt\ spivvidnoßennq
β β β β δ αik jl ij kl jl ik kl ij ml ij k
m
ml ib b b b g R R− + − = + jj k
m
(4)
ta
β β δ δij k ik j mj ik
m
mk ij
mb b, ,− = −Γ Γ , (5)
de tenzorni polq δΓ ij
h
ta δRijk
h
[ variaciqmy symvoliv Krystoffelq druho-
ho rodu ta tenzora Rimana, ” , ” — kovariantna poxidna na bazi metryçnoho
tenzora gij poverxni, Rijk
h
— komponenty tenzora kryvyny Rimana.
© I. V. POTAPENKO, 2010
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #2 199
200 I. V. POTAPENKO
Dovedennq. Nexaj poverxnq (1) zazna[ infinitezymal\no] deformaci] z vek-
torom zmiwennq y x x( , )1 2
. Qk vidomo [2, s. 66 – 68], variaci] koefici[ntiv per-
ßo] ta druho] kvadratyçnyx form poverxni, symvoliv Krystoffelq druhoho ro-
du ta tenzora Rimana [ takymy:
α δ β
β
β
βij ij i j j ig P g P g≡ = +. . , (6)
β δij ij i
m
m j i jb P b P≡ = + ∗
. , , (7)
δΓ ij
h
i j
h
i j
h h
ijP P b P b= − +∗ ∗
., , (8)
δ δ δRijk
h
ik j
h
ij k
h= −Γ Γ, , . (9)
Tut b b gj
h
ij
ih= .
PokaΩemo, wo umovy, zaznaçeni v teoremi, vykonano.
Varigvannqm rivnqn\ Haussa
b b b b g Rik jl ij kl ml ij k
m− = (10)
otrymu[mo (4).
I, analohiçno, varigvannqm rivnqn\ Majnardi – Petersona – Kodacci
b bij k ik j, ,− = 0 (11)
ma[mo (5).
Teoremu 1 dovedeno.
Zvyçajno, vynyka[ pytannq: çy [ vykonannq umov (4), (5) dostatnimy dlq is-
nuvannq vektora zmiwennq y x x( , )1 2
?
Teorema 2. Qkwo na rehulqrnij poverxni klasu C k
, k ≥ 3, isnugt\ dva sy-
metryçni tenzorni polq αij ta βij klasu C m
, m ≥ 2, qki zadovol\nqgt\
spivvidnoßennq (4), (5), de tenzorni polq δΓ ij
h
ta δRijk
h
vyznaçagt\sq za
formulamy
δ α α αΓ ij
h
mh
im j jm i ij m
g
= + −
2
( ), , , (12)
ta
δ δ δRijk
h
ik j
h
ij k
h= −Γ Γ, , , (13)
” , ” — kovariantna poxidna na bazi metryçnoho tenzora gij , to isnu[ infini-
tezymal\na deformaciq (2) z vektorom zmiwennq y x x( , )1 2
, dlq qko] ci ten-
zorni polq budut\ variaciqmy δgij ta δbij koefici[ntiv perßo] ta druho]
kvadratyçno] formy poverxni vidpovidno.
Dovedennq. Zaznaçymo, wo tenzorni polq (12), (13), qki vyznaçagt\sq çerez
zadane tenzorne pole αij , poky we ne [ variaciqmy niqkyx heometryçnyx
ob’[ktiv.
Dlq isnuvannq infinitezymal\no] deformaci] poverxni z vektorom zmiwen-
nqK(3) povynni vykonuvatys\ tak zvani umovy intehrovnosti [2, s. 103]
P P b P P bi j
h
i j
h
j i
h
j i
h
., .,− = −∗ ∗
, (14)
P P b P P bi j i
m
m j j i j
m
mi, . , .
∗ ∗+ = + . (15)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
NOVI RIVNQNNQ INFINITEZYMAL|NYX DEFORMACIJ POVERXON| V E3 201
PokaΩemo, wo pry isnuvanni na poverxni dvox symetryçnyx tenzornyx poliv
αij ta βij , qki zadovol\nqgt\ spivvidnoßennq (4), (5), isnu[ infinitezymal\na
deformaciq (2) z vektorom zmiwennq y x x( , )1 2
, dlq qko] ci tenzorni polq bu-
dut\ variaciqmy δgij ta δbij koefici[ntiv perßo] ta druho] kvadratyçno] for-
my poverxni, wo budut\ vyraΩatys\ za formulamy (6) ta (7) çerez komponenty
poxidno] vektora zmiwennq (3) vidpovidno.
Dlq c\oho, vykorystovugçy formulu (12), zapysu[mo (7), (8) takym çynom:
P P bi j ij i
m
m j, .
∗ = −β , (16)
P P b P bi j
h
ij
h
i j
h h
ij., = + −∗ ∗δΓ . (17)
Tobto my xoçemo vstanovyty umovy, za qkyx tenzorne pole βij bude variaci[g
koefici[ntiv druho] kvadratyçno] formy, a tenzorne pole (12) — variaci[g sym-
voliv Krystoffelq druhoho rodu pry deqkij infinitezymal\nij deformaci] po-
verxni z vektorom zmiwennq y x x( , )1 2
, çastynni poxidni qkoho magt\ vyh-
lqdK(3).
Oskil\ky tenzorni polq βij ta δΓ ij
h
symetryçni, to u vypadku isnuvannq
rozv’qzkiv systemy (16), (17), qka [ systemog typu Koßi vidnosno nevidomyx
funkcij Pi
h
. , Pi
∗
, spivvidnoßennq (14), (15), qki v svog çerhu [ dostatnimy dlq
isnuvannq vektora zmiwennq y x x( , )1 2
, budut\ vykonuvatys\.
Zapyßemo umovy intehrovnosti ci[] systemy. Dlq c\oho (16) kovariantno zdy-
ferencig[mo po xk
i, pidstavyvßy vyraz (17), otryma[mo
P P b P b b P bi j k ij k ik
m
i k
m m
ik mj i
m
, , .
∗ ∗ ∗= − + −( ) −β δΓ mmj k, . (18)
Proal\ternu[mo (18) i, zastosuvavßy totoΩnist\ Riççi, oderΩymo
P R b b Pm ijk
m
ij k ik j ik
m
m j ij
m
mk
m∗ ∗= − − + +. , ,β β δ δΓ Γ (( )b b b bik m j ij mk− –
– P b b b b P b bi k
m
m j j
m
mk i
m
m j k mk j
∗ −( ) − −. , ,( ) .
Vykorystovugçy formuly [2, s. 31, 34]
b b b bk
m
m j j
m
mk− = 0 , (19)
b bmj k mk j, ,= , (20)
R b b b bmij k ik m j ij mk= − (21)
ta oçevydnu rivnist\
P R P Rm ijk
m m
mij k
∗ ∗=. ,
perekonu[mosq, wo vykonano (5).
Analohiçno, kovariantno zdyferencigvavßy (17) po xk
, z vykorystan-
nqmK(16) budemo maty
P P b bi jk
h
ij k
h
ik i
m
mk j
h
., , .= + −( )δ βΓ +
+ P b g P b b P bi j k
h mh
mk m
l
l k ij
h
ij k
∗ ∗− −( ) −, . ,β . (22)
Proal\ternu[mo (22) i, zastosuvavßy totoΩnist\ Riççi, otryma[mo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, #2
202 I. V. POTAPENKO
P R P Rm
h
ij k
m
i
m
m jk
h
. . . .− = δ δ β βΓ Γij k
h
ik j
h
ik j
h
jk k
hb b, ,− + − –
– P b b b bi
m
mk j
h
m j k
h
. −( ) + P b b g b bi j k
h
k j
h mh
mk ij m j ik
∗ −( ) − −, , ( )β β +
+ g P b b b b P b bmh
m
l
l k ij l j ik
h
ij k ik j. , ,( ) ( )− − −∗ . (23)
Vykorystavßy formuly (13), (20), a takoΩ
R b b b bij k
h
j
h
ik k
h
ij. = − ,
(23) perepyßemo u vyhlqdi
δ β β β βR b b g b b Pijk
h
ik j
h
jk k
h mh
mk ij m j ik m. ( )= − − − − .. . .
h
ij k
m mh
m
l
lij kR g P R− . (24)
Vykorystavßy matematyçnu operacig opuskannq indeksu h v (24) za dopomohog
metryçnoho tenzora ghn , otryma[mo
g R b b bhn ij k
h
ik jn jk kn nk ijδ β β β. = − − + βnj ik hn m
h
ij k
m
n
l
lij kb g P R P R− −. . . . (25)
Vraxovugçy formulu
R g Rlij k lm ij k
m= .
ta (6), perekonu[mos\, wo (25) zbiha[t\sq z (4).
TeoremuK2 dovedeno.
TeoremaK2 pokazu[, wo umovy (4), (5), qki v teori] infinitezymal\nyx defor-
macij [ pevnymy analohamy rivnqn\ Haussa ta rivnqn\ Petersona – Majnardi –
Kodacci v klasyçnij teori] poverxon\, [ ne lyße neobxidnymy dlq isnuvannq in-
finitezymal\no] deformaci], a j dostatnimy.
Stosovno vidnovlennq polq deformacij y x x( , )1 2
za vidomymy funkciqmy
αij ta βij ma[mo try rivnqnnq dlq ßesty funkcij, adΩe kil\kist\ sutt[vyx
rivnqn\ u spivvidnoßennqx (4), (5) dorivng[ tr\om. U spivvidnoßennqx (14), (15)
cq kil\kist\ takoΩ dorivngvala tr\om rivnqnnqm dlq ßesty funkcij Pi
h
. , Pi
∗
.
Takym çynom, qk vydno z teoremyK2, spivvidnoßennq (4), (5) moΩna vvaΩaty no-
vog formog umov intehrovnosti isnuvannq vektora zmiwennq pry infinitezy-
mal\nij deformaci] poverxni v E3 .
Suçasni doslidΩennq infinitezymal\nyx deformacij poverxon\ u tryvymir-
nomu evklidovomu prostori [3 – 5] potrebugt\ al\ternatyvnyx pidxodiv do ]x vy-
vçennq. Otrymanyj rezul\tat dast\ zmohu doslidΩuvaty rizni, bil\ß skladni
typy infinitezymal\nyx deformacij poverxon\ v E3 .
1. Bonnet O. J. Memoire sur la theorie des surfaces applicables sur une surface donee // J. Ěcole
Polytechnique. – 1867. – 25. – P. 1 – 51.
2. Bezkorovajna L. L. Areal\ni neskinçenno mali deformaci] i vrivnovaΩeni stany pruΩno]
obolonky. – Odesa: AstroPrynt, 1999. – 168 s.
3. Lejko S. H., Fedçenko G. S. Infinitezymal\ni povorotni deformaci] poverxon\ ta ]x zasto-
suvannq v teori] pruΩnyx obolonok // Ukr. mat. Ωurn. – 2003. – 55, # 12. – S. 1697 – 1703.
4. Fomenko V. T. ARG-deformations of a hypersurface with a boundary in Riemannian space //
Tensor. – 1993. – 54. – P. 28 – 34.
5. Ferapontov E. V. Surfaces in 3-spaces possessing non trivial deformations which preserve the shape
operator // Different. Geometry and Integrable Systems in Differential Geometry (Tokyo, July 17 –
21, 2001). – Providence (R. I.): Amer. Math. Soc., 2002. – P. 145 – 159.
OderΩano 22.09.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
|
| id | umjimathkievua-article-2856 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:31:37Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2e/be2ff82c17a1bc9dcd08e64fd7f0952e.pdf |
| spelling | umjimathkievua-article-28562020-03-18T19:39:03Z New equations of infinitesimal deformations of surfaces in $E_3$ Нові рівняння інфінітезимальних деформацій поверхонь в $E_3$ Potapenko, I. V. Потапенко, І. В. We establish a condition for two symmetric tensor fields that is necessary and sufficient for the existence of a displacement vector in the case of infinitesimal deformation of a surface in the Euclidean space E 3. Доказано необходимое и достаточное условие, которому должны удовлетворять два симметрических тензорных поля для существования вектора смещения при инфинитезимальной деформации поверхности в евклидовом пространстве $E_3$. Institute of Mathematics, NAS of Ukraine 2010-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2856 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 2 (2010); 199–202 Український математичний журнал; Том 62 № 2 (2010); 199–202 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2856/2464 https://umj.imath.kiev.ua/index.php/umj/article/view/2856/2465 Copyright (c) 2010 Potapenko I. V. |
| spellingShingle | Potapenko, I. V. Потапенко, І. В. New equations of infinitesimal deformations of surfaces in $E_3$ |
| title | New equations of infinitesimal deformations of surfaces in $E_3$ |
| title_alt | Нові рівняння інфінітезимальних деформацій поверхонь в $E_3$ |
| title_full | New equations of infinitesimal deformations of surfaces in $E_3$ |
| title_fullStr | New equations of infinitesimal deformations of surfaces in $E_3$ |
| title_full_unstemmed | New equations of infinitesimal deformations of surfaces in $E_3$ |
| title_short | New equations of infinitesimal deformations of surfaces in $E_3$ |
| title_sort | new equations of infinitesimal deformations of surfaces in $e_3$ |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2856 |
| work_keys_str_mv | AT potapenkoiv newequationsofinfinitesimaldeformationsofsurfacesine3 AT potapenkoív newequationsofinfinitesimaldeformationsofsurfacesine3 AT potapenkoiv novírívnânnâínfínítezimalʹnihdeformacíjpoverhonʹve3 AT potapenkoív novírívnânnâínfínítezimalʹnihdeformacíjpoverhonʹve3 |