
frobanius manifold

vibhedak jyaamiti key ganitiya kshetra mein, ack frobanius manifold, jise dubrovin dwara pesh kiya gayaa haye, sparshrekha sthaan parr ack nishchit sangat gunak sanrachana key saath ack sapaat remanian manifold haye. yeh avadharana frobanius beejganit kii dhaarana koo sparshrekha bandalon mein samanya karati haye.
frobanius manifolds swabhavik roop seey symplectic topology key vishay mein hotey hain, vishesh roop seey quantum cohomology. sabse vyaapak paribhasha remanian supermanifolds kii shreni mein haye. humm yahaan charcha koo sucharu (vaastavik) kaee gunaa taka seemit curr denge. jatil manifolds parr pratibandh bhee sambhav haye.
paribhasha.
m" koo ack chikna manifold honey dein. m" parr ack "affine flat" sanrachana vector rikt sthaan kaa ack sheef "t" "f" haye joe bindu roop seey "tm" spain kartaa haye aur isake khandon key jode kaa sparshak koshthak gayab hoe jaataa haye.
ack sthaaniya udaaharan key roop mein "m" key chart parr samanvaya sadish kshetron parr vichaar karein. ack manifold ack affine sapaat sanrachana koo sweekaar kartaa haye yadi koee chart key ack aavaran parivaar key liye aise vectorfield koo ack saath chipka sakata haye.
aage "m" parr ack remanian metrick "g" diyaa jaae. yeh sapaat sanrachana key liye sangat haye yadi "g" ("x", "y") sabhi sapaat sadish kshetron "x" aur "y" key liye sthaaniya roop seey sthir haye.
ack remanian manifold ack sangat samatulya sanrachana koo sweekaar kartaa haye yadi aur kewal tabhi jaba isaki vakrata tensor harr jagah gayab hoe jaae.
tii. aim.". parr "commutative utpaadon *" kaa ack parivaar "aiss" 2 (tii * "aim") key ack khand "aeye" key barabar haye.
hamein sampatti key atirikt kii aavashyakta haye
isaliye, rachana "g" #.65 "a" ack samamit 3-tensor haye.
iska vishesh roop seey tatparya haye kii sthir utpaad key saath ack raikhik frobanius manifold ("m", "g", *) ack frobanius beejganit "m" haye.
diyaa gayaa ("g", "t" "f", "a"), ack "sthaaniya vibhav φ" ack sthaaniya chikna falan haye jaise kii
sabhi sapaat sadish kshetron "x", "y" aur "z" key liye.
ack "frobanius manifold" ("aim", "jee", "*) abb samamit 3-tensor" aeye "key saath ack sapaat remanian manifold (" aim "," jee ") haye joe harr jagah ack sthaaniya kshamata koo sweekaar kartaa haye aur sahayogi haye.
praathamik guna.
utpaad kii sahakriyashilta * sthaaniya kshamata "φ" mein nimnalikhit dwighaat pde key barabar haye."
jahan einstein kaa yog paripati nihit haye, φ, ack nirdeshank sadish kshetra shna/shna "x" a "dwara falan φ key aanshik vyutpann koo darshaata haye joe sabhi koo sapaat maanaa jaataa haye. "g" "ef" "metrick key vyutkram key gunank hain.
isaliye sameekaran koo saha-sakriyata sameekaran yaa vitten-disgraph-verlinde-verlinde (doubleu. dee. vee. vee.) sameekaran kahaa jaataa haye.
udaaharan.
frobanius algebras key alawa, quantum cohomology seey udaaharan utpann hotey hain. arthaat, ack ardh-sakaratmak samroopata ("m", "ω") koo dekhate hue, isake summ quantum samroopata mein 0 kaa ack khula pados "u" maujood haye, jisamen c parr novikov ring ("m", "ω") haye, jisase kii "u" mein "a" key liye badaa quantum utpaad * "a" vishleshanatmak haye. abb pratichhedan roop "g" = <·,·> key saath "u" ack (jatil) frobanius manifold haye.
frobanius manifolds key udaharanon kaa doosara badaa varg ekatva siddhant seey aataa haye. arthaat, ack prithak vilakshanta key laghu viroopan key sthaan mein ack frobanius manifold sanrachana hoti haye. yeh frobanius manifold sanrachana kyoji saito key adim roopon seey bhee sambandhit haye.
sandarbh.
 2. yuu.aaee. manin, aiss.aeye. markulovah "ardh-saral frobanius (super) manifolds aur quantum cohomology off pee. aar", topol. gair-raikhik vishleshan mein vidhiyaan 9 (1997), pp. 107-161

