
baar-baar honey vaalaa tensor

ganit aur bhautiki mein, ack baar-baar aaney vaalaa tensor, ack manifold "m" parr ack connection sootra _ 1 key sambandh mein, ack tensor "t" haye jisake liye "m" parr ack-roop "ω" haye jaise kii
udaaharan.
samanantar tensor.
aavarti tensor key liye ack udaaharan samanantar tensor hain jinhen paribhaashit kiya gayaa haye 
kuchh connection sootra _ 1 key sambandh mein.
yadi humm ack chhadm-remanian manifold sootra _ 5 lete hain tou metrick "jee" ack samanantar haye aur isaliye isake levi-sivita connection key sambandh mein aavarti tensor haye, jise isake madhyam seey paribhaashit kiya jaataa haye.
aur isaki sampatti koo marod-mukt honah chaahiye.
samanantar sadish kshetra (sootra _ 7) aavarti tensor key udaaharan hain joe ganitiya anusandhaan mein mahatva paate hain. udaaharan key liye, yadi sootra _ 8 ack chhadm-remanian manifold santoshjanak parr ack aavarti gair-shoonya sadish kshetra haye
kuchh bund ack-roop sootra _ 10 key liye, tou x koo ack samanantar sadish kshetra mein punah praapt kiya jaa sakata haye. vishesh roop seey, gair-samanantar aavarti sadish kshetra null sadish kshetra hain.
metrick sthaan.
ack aur udaaharan wheel sanrachanaon key sambandh mein dikhaai detaa haye. aitihaasik roop seey, vektaron key samanantar parivahan key gunon aur unaki lambai key sambandh mein herman weill key vichaaron seey weill sanrachanaaen ubhari hain. yeh maang karke kii ack manifold mein ack samaantar parivahan iss tarah seey hoe kii manifold sthaaniya roop seey ack samaantar sthaan hoe, yeh dikhaaya gayaa kii prerit connection mein ack lupt hoe rhaa marod tensor thaa.
isake alawa, unhone daavaa kiya kii kaee gunaa mein ack vishesh samanantar parivahan honah chaahiye jisamen dou parivahan vactors kaa anupaat nishchit haye. sambandhit connection sootra _ 12 joe iss tarah key samanantar parivahan koo santusht kartaa haye
kuchh ack-roop sootra _ 14 key liye. iss tarah kaa metrick sootra _ 12 key sambandh mein ack aavarti tensor haye. parinamswarup, wheel koo parinami manifold sootra _ 5 kahaa jaataa haye jisamen affine connection sootra _ 1 aur aavarti metrick sootra _ 18 ack metrick sthaan hotaa haye. iss arth mein, wheel kewal ack metrick kaa ullekh naheen curr rhaa thaa, balki sootra _ 18 dwara paribhaashit anuroop sanrachana kaa ullekh curr rhaa thaa.
anuroop parivartan sootra _ 20 key tahat, formula _ 14 formula _ 22 key roop mein parivartit hotaa haye. yeh sootra _ 24 parr ack canonical mapp sootra _ 23 koo prerit kartaa haye jise paribhaashit kiya gayaa haye
jahan sootra _ 26 anuroop sanrachana haye. sootra _ 27 koo ack wheel sanrachana kahaa jaataa haye, jise aam taur parr sampatti key saath maanchitra key roop mein paribhaashit kiya jaataa haye.
baar-baar antariksh samay.
ack aavarti tensor kaa ack aur udaaharan aavarti spacetime parr vakrata tensor sootra _ 29 haye, jisake liye