Tenzor Rimana R i j k s displaystyle R ijk s tenzor vnutrishnoyi krivini mnogovida z yavlyayetsya pri rozglyadi komutatora kovariantnih pohidnih kovariantnogo vektora divitsya stattyu Diferencialna geometriya 1 j k a i R i j k s a s displaystyle 1 qquad nabla j nabla k a i R ijk s a s Zmist 1 Deyaki totozhnosti 2 Isnuvannya dekartovoyi sistemi koordinat 3 Poglyad iz ohoplyuyuchogo evklidovogo prostoru 4 Div takozh 5 LiteraturaDeyaki totozhnosti Redaguvati 2 R i j k s j G i k s k G i j s G i j p G p k s G i k p G p j s displaystyle 2 qquad R ijk s partial j Gamma ik s partial k Gamma ij s Gamma ij p Gamma pk s Gamma ik p Gamma pj s nbsp Zamist kovariantnih komponent a i displaystyle a i nbsp mozhna pidstaviti bazisni vektori r i displaystyle mathbf r i nbsp 3 j k r i R i j k s r s displaystyle 3 qquad nabla j nabla k mathbf r i R ijk s mathbf r s nbsp I vrahovuyuchi sho kovariantna pohidna vid bazisnih vektoriv j r i displaystyle nabla j mathbf r i nbsp dorivnyuye vektoram povnoyi krivini b i j displaystyle mathbf b ij nbsp divitsya Prosti obchislennya diferencialnoyi geometriyi mayemo 3 j b i k k b i j R i j k s r s displaystyle 3 qquad nabla j mathbf b ik nabla k mathbf b ij R ijk s mathbf r s nbsp Domnozhimo formulu 3 skalyarno na r p displaystyle mathbf r p nbsp i vrahuyemo ortogonalnist vektoriv krivini do mnogovidu r s b i j 0 displaystyle mathbf r s cdot mathbf b ij 0 nbsp V rezultati oderzhuyemo formulu dlya kovariantnih komponent tenzora Rimana R p i j k R i j k s r p r s r p j b i k r p k b i j displaystyle R pijk R ijk s mathbf r p cdot mathbf r s mathbf r p cdot nabla j mathbf b ik mathbf r p cdot nabla k mathbf b ij nbsp j r p b i k j r p b i k k r p b i j k r p b i j displaystyle nabla j mathbf r p cdot mathbf b ik nabla j mathbf r p cdot mathbf b ik nabla k mathbf r p cdot mathbf b ij nabla k mathbf r p cdot mathbf b ij nbsp 0 b j p b i k 0 b k p b i j b p j b i k b p k b i j displaystyle 0 mathbf b jp cdot mathbf b ik 0 mathbf b kp cdot mathbf b ij mathbf b pj cdot mathbf b ik mathbf b pk cdot mathbf b ij nbsp abo pislya zmini znaku i perejmenuvannya indeksiv 4 R i j k l b i k b j l b i l b j k displaystyle 4 qquad R ijkl mathbf b ik cdot mathbf b jl mathbf b il cdot mathbf b jk nbsp Yak mozhna pobachiti z ostannogo rivnyannya v skalyarnih dobutkah indeksi k displaystyle k nbsp i l displaystyle l nbsp perestavleni tenzor Rimana antisimetrichnij za pershoyu paroyu indeksiv i j displaystyle ij nbsp i za drugoyu paroyu indeksiv k l displaystyle kl nbsp pri perestanovci zmenshuvane i vid yemnik u pravij chastini formuli 4 minyayutsya miscyami 5 R i j k l R j i k l R i j l k displaystyle 5 qquad R ijkl R jikl R ijlk nbsp Takozh legko bachiti sho tenzor Rimana ne zminyuyetsya pri perestanovci pershoyi pari indeksiv i j displaystyle ij nbsp z drugoyu paroyu indeksiv k l displaystyle kl nbsp pri perestanovci u mnozhnikah zmenshuvanogo indeksi perestavlyayutsya ale oskilki velichini b i j displaystyle mathbf b ij nbsp simetrichni za indeksami to skalyarnij dobutok zmenshuvanogo ne zminitsya u vid yemniku analogichno ale spivmnozhniki v skalyarnomu dobutku minyayutsya miscyami sho ne vplivaye na rezultat 6 R i j k l R k l i j displaystyle 6 qquad R ijkl R klij nbsp Zgortka tenzora Rimana za pershim i tretim indeksami abo sho ekvivalentno za drugim i chetvertim indeksami daye simetrichnij tenzor drugogo rangu R i k displaystyle R ik nbsp yakij nazivayetsya tenzorom Richchi 7 R i k g j l R i j k l displaystyle 7 qquad R ik g jl R ijkl nbsp Tenzor Richchi simetrichnij 8 R i k R k i displaystyle 8 qquad R ik R ki nbsp Tenzor Richchi mozhna takozh zgornuti za indeksami oderzhavshi skalyarnu krivinu 9 R g i k R i k displaystyle 9 qquad R g ik R ik nbsp Vrahovuyuchi 4 mayemo 10 R b i i 2 b i j b j i displaystyle 10 qquad R mathbf b i i 2 mathbf b i j cdot mathbf b j i nbsp Komutator dlya kontravariantnogo vekora oderzhuyemo pidnyavshi indeks i displaystyle i nbsp u formuli 1 11 j k a i R j k s i a s R j k i s a s R s j k i a s displaystyle 11 qquad nabla j nabla k a i R jk si a s R jk is a s R sjk i a s nbsp Oskilki komutator kovariantnih pohidnih i j displaystyle nabla i nabla j nbsp diye na dobutok tenzoriv T U displaystyle TU nbsp za pravilom diferencialnogo operatora i j T U i j T U T i j U displaystyle nabla i nabla j TU nabla i nabla j T U T nabla i nabla j U nbsp to mi mozhemo koristuyuchis formulami 1 i 11 obchisliti diyu komutatora kovariantnih pohidnih na tenzor yakij ye dobutkom vektoriv Ale dovilnij tenzor mozhna predstaviti linijnoyu kombinaciyeyu takih elementarnih tenzoriv tomu pri diyi komutatora na dovilnij tenzor z bud yakoyu kilkistyu verhnih ta nizhnih indeksiv mayemo 12 j k T l 1 l 2 i 1 i 2 R s j k i 1 T l 1 l 2 s i 2 R s j k i 2 T l 1 l 2 i 1 s R l 1 j k s T s l 2 i 1 i 2 R l 2 j k s T l 1 s i 1 i 2 displaystyle 12 qquad nabla j nabla k T l 1 l 2 cdots i 1 i 2 cdots R sjk i 1 T l 1 l 2 cdots si 2 cdots R sjk i 2 T l 1 l 2 cdots i 1 s cdots cdots R l 1 jk s T sl 2 cdots i 1 i 2 cdots R l 2 jk s T l 1 s cdots i 1 i 2 cdots nbsp Tenzor Rimana zadovolnyaye dvi totozhnosti Bianki Algebrayichna totozhnist Bianki ciklichna perestanovka indeksiv i j k displaystyle ijk nbsp 13 R s i j k R s j k i R s k i j 0 displaystyle 13 qquad R sijk R sjki R skij 0 nbsp Diferencialna totozhnist Bianki ciklichna perestanovka indeksiv p j k displaystyle pjk nbsp 14 p R i j k s j R i k p s k R i p j s 0 displaystyle 14 qquad nabla p R ijk s nabla j R ikp s nabla k R ipj s 0 nbsp Isnuvannya dekartovoyi sistemi koordinat RedaguvatiYaksho isnuye dekartova sistema koordinat to R i j k l 0 displaystyle R ijkl 0 nbsp Yaksho na mnogovidi isnuye dekartova sistema koordinat v yakij metrichnij tenzor dorivnyuye odinichnij matrici g i j d i j displaystyle g ij delta ij nbsp to v cij sistemi koordinat vsi pohidni metrichnogo tenzora k g i j 0 displaystyle partial k g ij 0 nbsp a otzhe i vsi simvoli Kristofelya totozhno dorivnyuyut nulyu G i j k 1 2 g k s i g j s j g i s s g i j 0 displaystyle Gamma ij k 1 over 2 g ks partial i g js partial j g is partial s g ij 0 nbsp Otzhe i vsi komponenti tenzora Rimana v dekartovij sistemi koordinat dorivnyuyut nulyu R i j k s j G i k s k G i j k G j p s G i k p G k p s G i j p 0 displaystyle R ijk s partial j Gamma ik s partial k Gamma ij k Gamma jp s Gamma ik p Gamma kp s Gamma ij p 0 nbsp Ale oskilki tenzor Rimana pri perehodi v inshu sistemu koordinat peretvoryuyetsya po tenzornim pravilam R i j k s R l m n p u s u p u l u i u m u j u n u k displaystyle tilde R ijk s R lmn p partial tilde u s over partial u p partial u l over partial tilde u i partial u m over partial tilde u j partial u n over partial tilde u k nbsp to vin dorivnyuye nulyu v bud yakij inshij sistemi koordinat na comu mnogovidi Yaksho R i j k l 0 displaystyle R ijkl 0 nbsp to mozhna pobuduvati dekartovu sistemu koordinatNehaj tenzor Rimana totozhno dorivnyuye nulyu v deyakij zv yaznij oblasti mnogovida Vizmemo dovilnu tochku P displaystyle P nbsp v mezhah ciyeyi oblasti cya tochka bude pochatkom nashoyi majbutnoyi dekartovoyi sistemi koordinat V tochci viberemo yakijs ortonormovanij bazis vektori cogo bazisu budut zadavati dodatni napryamki koordinatnih osej majbutnoyi sistemi koordinat Rozglyanemo odin iz vektoriv bazisu yakij poki sho dlya prostoti poznachimo bukvoyu v displaystyle v nbsp vzagali to kilkist bazisnih vektoriv n displaystyle n nbsp i treba bulo b poznachiti indeksom yakij iz bazisnih vektoriv mi rozglyadayemo ale poki mi zoseredimosya na pobudovi odniyeyi koordinati Koristuyuchis paralelnim perenesennyam pochinayuchi z tochki P displaystyle P nbsp v kozhnij tochci oblasti mnogovida pobuduyemo vektor paralelnij vektoru v displaystyle v nbsp Rezultat perenesennya ne zalezhit vid shlyahu perenosu oskilki tenzor Rimana dorivnyuye nulyu a zalezhit tilki vid kincevoyi tochki Takim chinom mi oderzhali v nashij oblasti vektorne pole 1 v i v i u 1 u 2 u n displaystyle 1 qquad v i v i u 1 u 2 dots u n nbsp yake do togo zh ye postijnim stosovno kovariantnogo diferenciyuvannya tobto spravedlivi rivnosti 2 j v i 0 displaystyle 2 qquad nabla j v i 0 nbsp Z ostannogo rivnyannya vrahovuyuchi oznachennya kovariantnoyi pohidnoyi i simetriyu simvoliv Kristofelya znahodimo 3 0 j v i i v j j v i G j i k v k i v j G i j k v k j v i i v j 0 displaystyle 3 qquad 0 nabla j v i nabla i v j partial j v i Gamma ji k v k partial i v j Gamma ij k v k partial j v i partial i v j 0 nbsp Teper oskilki 4 j v i i v j displaystyle 4 qquad partial j v i partial i v j nbsp To vektor ye gradiyentom deyakoyi skalyarnoyi funkciyi ϕ ϕ u 1 u 2 u n displaystyle phi phi u 1 u 2 dots u n nbsp 5 v i i ϕ displaystyle 5 qquad v i partial i phi nbsp Funkciyu ϕ displaystyle phi nbsp v yakijs tochci Q displaystyle Q nbsp oblasti mnogovida mozhna obchisliti cherez integral po krivij sho spoluchaye pochatok koordinat P displaystyle P nbsp i tochku Q displaystyle Q nbsp 7 ϕ Q P Q v i d u i displaystyle 7 qquad phi big Q int P Q v i du i nbsp prichomu rezultat integruvannya ne zalezhit vid krivoyi vnaslidok formuli Stoksa i rivnosti 5 Funkciya ϕ ϕ u 1 u 2 u n displaystyle phi phi u 1 u 2 dots u n nbsp i bude odniyeyu z koordinat Teper povernemosya do inshih vektoriv bazisu cogo razu uzhe pronumeruyemo ci vektori indeksom vzyatim u duzhki Tak samo dlya kozhnogo takogo vektora pobuduyemo v nashij oblasti vidpovidne postijne vektorne pole yake ye gradiyentom vidpovidnoyi koordinati 8 v i k i ϕ k displaystyle 8 qquad v i k partial i phi k nbsp Oskilki paralelne perenesennya grupi vektoriv zberigaye skalyarni dobutki mizh nimi a v pochatku koordinat ci skalyarni dobutki dorivnyuyut odinichnij matrici to v usij oblasti mayemo 9 g i j ϕ k u i ϕ l u j d k l displaystyle 9 qquad g ij partial phi k over partial u i partial phi l over partial u j delta kl nbsp tobto koordinati ϕ k displaystyle phi k nbsp ye dekartovimi Poglyad iz ohoplyuyuchogo evklidovogo prostoru RedaguvatiRozglyanemo rivnist 10 R i j k l b i k b j l b i l b j k 0 displaystyle 10 qquad R ijkl mathbf b ik cdot mathbf b jl mathbf b il cdot mathbf b jk 0 nbsp v yakijs tochci P displaystyle P nbsp mnogovidu i dvi geodezichni liniyi sho prohodyat cherez cyu tochku ale v riznih napryamkah Krivini cih geodezichnih dorivnyuyut k b i j t i t j displaystyle mathbf k mathbf b ij tau i tau j nbsp k b i j t i t j displaystyle tilde mathbf k mathbf b ij tilde tau i tilde tau j nbsp Teper domnozhimo 10 na dobutok t i t j t k t l displaystyle tau i tilde tau j tau k tilde tau l nbsp oderzhimo k k b i j t i t j 2 0 displaystyle mathbf k cdot tilde mathbf k mathbf b ij tau i tilde tau j 2 geq 0 nbsp Visnovok krivini vsih geodezichnih napryamleni priblizno v odin bik mnogovid ne maye sidlovih tochok v yakih bi rizni geodezichni vikrivlyalisya v protilezhni boki Div takozh RedaguvatiRozklad Richchi Peretvorennya krivini Tenzor kruchennyaLiteratura RedaguvatiVolodimir Kiosak Oleksandr Prishlyak Rimanova geometriya Kiyiv mehmat KNU 2017 49 s John M Lee en 2018 Introduction to Riemannian Manifolds Springer Verlag ISBN 978 3 319 91754 2 Otrimano z https uk wikipedia org w index php title Tenzor krivini amp oldid 38991846
