ON THE STRUCTURE OF THE EULER MAPPING

Transcription

1 Electocal tacpto Mathematcal Ittute, Slea Uet Opaa, Cech Republc Mach Th tet a electoc tacpto o the ogal eeach pape D. upa, O the tuctue o the Eule mappg, Ach. Math., Scpta Fac. Sc. Nat. UJEP Bue, X: 55-6, 974 Tpogaphcal eo hae bee coected. ARCH. MATH., SCRIPTA FAC. SCI. NAT. UJEP BRUNENSIS, X: 55-6, 974 ON THE STRUCTURE OF THE EUER MAPPING Demete upa, Bo (Receed Octobe 4, 97). Itoducto et R, R m be eal Eucldea pace o dmeo, m, m epectel, ( R, R ) the ecto pace o all lea mappg om R to R m, m ( R, R ) the ecto pace o all mmetc blea mappg om R to R m, m UR ad VR ome ope et. We wte R= R. Put m m = U V ( R, R ), = U V ( R, R ) ( R, R ) (the catea poduct), ad code ad a deetable maold wth atual coodate (,, ) ad (,,, ) epectel (, m). Deote b Γ the et o all deetable map : U V (a, o cla C ), ad wte D o the -th deate o the map [], =,. Aume that we hae a eal ucto o ad a compact doma ΩU. The data ge e to the eal ucto () Γ (, ( ), D( )) dr Ω (wth d = d d ) whch o pcpal teet aou poblem o the calculu o aato (ee e.g. [3]). The etemal aocated wth ae the deed a oluto Γ o the o called Eule equato m () ( ) = =, =,,, m.

2 Hee, eewhee th pape, the uual ummato coeto ued. The epeo ( ) deed b () ae ucto o. Put = d d, ad dee a -om ( ) o b the omula ( ) ( ). = It ca be eal checed that the -om ( ) depedet o the patcula choce o coodate o. We hall call each ucto o the agage ucto, ad the -om ( ) the Eule om aocated wth the agage ucto. The ecto pace (oe R) o all agage ucto deoted b ( ), ad the ecto pace o all -om o (oe R) deoted b Ω ( ). Ceta ucet codto o the detcal ahg o the let-had de o the Eule equato (), o, whch the ame, o ( ) =, ae ow ad equetl ued aou calculato. Suppoe that ( ) o the om o the o called degece epeo (3) = ƒ + ƒ, whee,, ae ome ucto o U V. The we ee at oce that ( ) =. It alo ow that o the cae m = codto (3) ecea: th a clacal popoto o Couat ad Hlbet []. We meto ut two cae whe codto (3) ued:. I the clacal mechac [5] ad the geeal elatt [6], (3) ee o eplacg the ge agage ucto b a moe mple oe.. I the theo o aat aatoal poblem, o deto o the o called geealed aat taomato [8] (ee alo [4], [9]). O the othe had, a complete decpto o the agage ucto atg ( ) = ha ot et bee ge ule m =. The goal o th pape to ge uch a decpto. I othe wod we hall tud the eel o the lea mappg ( ) ( ) Ω ( ) whch wll be eeed to a the Eule mappg.. Deto ad lemma. Fo the pupoe o th pape t uce to dee what we mea b hootal deetal om o the catea poduct o ope ubet o Eucldea pace. et V ad W be ome ope et the te dmeoal Eucldea pace R ad R m, ad code the catea poduct V W ad the atual poecto :V W V o the t acto. A taget ecto at a pot (, w) V W called -etcal, D(, w) =.

3 A deetal om o V W called -hootal t ahe wheee oe o t agumet (.e. taget ecto) a -etcal ecto. et u tu to the otato o Itoducto. We deote : U, : U V, : U V the atual poecto ad hall theeoe pea about -hootal, -hootal, ad -hootal deetal om. Coepodgl, we hall wte Ω U ( ), Ω U ( ) V, ad Ω U V ( ) o the pace o all -hootal -om, -hootal -om, ad -hootal -om (emembe that = dm U). Notce that the Eule om, ( ), a elemet o Ω U V( ). et Γ dee the mappg U ( ) = (, ( ), D ( )) ad deote b the coepodg mappg duced o deetal om o. Thu, a deetal p-om o, the a deetal p-om o U. emma. Thee oe ad ol oe mappg Ω U V( ) h( ) ΩU ( ) atg the ollowg two codto:. h lea oe the g o ucto o. I Ω U V ( ) a abta -om, the = h( ) o all Γ. Poo. I the mappg h et, t oboul uque. et be a abta elemet o Ω U ( ) V. I the atual coodate (,, ), ha the epeo (4) = gdd + g d d! =,, d d d d + ( whch the ucto g ae uppoed to be atmmetc all ubcpt), the we dee (5) h( ) = g + g d d =,, It mmedatel clea that codto ad ae ated. emma. The mappg h uecte. Poo. et (6) = d d be a abta -hootal -om. We tae

4 whee = d d +, = dd d d d+ d ( ). The equalt h( ) = ollow om (5). We ote that the om om the poo aat ude coodate taomato o. It ha bee toduced, a pecal cae, b Satc [7] coecto wth ome geometc codeato coceg the tuctue o the calculu o aato. I ode to hote the poo o ou ma theoem we tate the ollowg eplct omula o the eteo deetal d o a om Ω U ( ) V. emma 3. et Ω U ( ) V be epeed a (4). The d epeed a (7) g g d = d d d + =,, g g g g,, g g d d d d + + g ( + )! + g g ( + )! d d d g dd d + d d d g + d! =,, d d d d d d +. Poo. The omula ollow b a taghtowad calculato. 3. The eel o the Eule mappg. The ma eult o th wo cotaed the ollowg: Theoem. et ( ) be a agage ucto. The the ollowg two codto ae equalet:. The Eule om aocated wth ahe, ( ) =.. Thee et a -om Ω U V ( ) uch that a) h( ) = d d, b) d =.

5 The -om uquel detemed b. Poo. Suppoe that ( ) =. The the elato () hold o all (,,, ), ad ae equalet wth the tem (8) + =, (9) =. Fom the t codto (8) we d that mut be o the om () = + =,,, whee ad do ot deped o ad ae atmmetc,,. et u eame the ecod codto (9). Ate ome calculato we get () + = <,, > +,, =. Sce the coecet at do ot deped o the mut ah epaatel. I th wa we hae obtaed that ate ( ) =, the o the om () ad the codto () ae ated. We aet that the ucto, ae uque: t ollow om () that = = = +, ( ),,,

6 = Coequetl, we put = < <,, = d d +! =,, d d d d d d + we obta, b (5) ad emma 3, that codto om the theoem ated b. At the ame tme we hae poed that the -om uque. Coeel, uppoe that we hae a -om Ω U V ( ) atg. B compao wth emma 3 t ca be ee at oce that the agage ucto deed b a) ate the codto. Th poe the Theoem. Rema. et ( ) be a agage ucto atg the codto ( ) =, ad the coepodg -om om the Theoem. Sce the ucto, do ot deped o, the om ca be egaded a deed o U V. The popet d = the mea that we ca d, at leat locall, a ( ) -om o U V uch that = d. (Th ollow om the well-ow Pocae lemma coceg the o called cloed om.) We thu obee that ate the elato () d d = h ( d ). Coeel, we tae a abta ( ) -om deed o U V ad dee b elato () we ca ee at oce that the ucto lead to the equalt ( ) =. Thu, hag md emma, we ca a that codto () wth abta ( ) -om o U V decbe all the agage ucto o whch ( ) =. Rema. We ote that all codeato om th pape ca be eteded to the cae whe thee ge a bed maold ( Y,, X), ad agage ucto deed o the t et pologato o the bed maold ae codeed (ee [4] ad [8]). Rema 3. a) I =, the -om o U V ae ut eal ucto. I we wte (,,«) o the atual coodate o th cae we get, o a abta ucto F o U V, ad F hdf ( ) = F + «d,.

7 F = + F «. b) I =, the the geeal -om o U V ca be epeed a = d + gd. Ate ome calculato g g hd ( ) = + + d d. I th omula = =, = =. The agage ucto leadg to eo Eule om hae theeoe to be o the om g g = + +. c) I geeal, oe ca poceed the ame mae a the cae a) o b). Reeece [] Couat, R., D. Hlbet, Method o mathematcal phc, Vol. I, New Yo (953) [] Deudoe, J., Foudato o mode aal, New Yo (96) [3] Gelad, I.M., S. Fom, Calculu o aato, New Jee (967) [4] upa, D., agage theo beed maold, Rep. Math. Ph. (97), -33 [5] adau,.d., E.M. ht, Mechac, Pegamo Pe (969) [6] Mcech, N.V., Phcal eld the geeal elatt theo, Mocow 969 (Rua) [7] Satc, J., O the geometc tuctue o clacal eld theo agaga omulato, Poc.Camb. Phl. Soc. (97), 68, [8] Tautma, A., Noethe equato ad coeato law, Commu. Math. Ph. 6 (967), 48-6 [9] Tautma, A., Coeato law geeal elatt, : Gatato, New Yo (96) D. upa 6 37 Bo, otlaa Cecholoaa