<<

c Amitabha Lahiri: Lecture Notes on Differential for Physicists 2011 iesoa dimensional • attention are our we hold continuity, will • and that sets course. object open this fundamental during of the notions define to the ready have we that Now 2 Chapter ihaoet-n a nosm pnstof set open some onto map one-to-one a with thn.I practice, In hand. at means Smooth differentiable. times a of that C • iii ∞ ii i hti,eeypito aiodhsa pnneighbourhood open an has manifold a of every is, That ehv endsot aiod.Amr eea ento is definition general more A manifolds. smooth defined have We . ) can We ) I n intersection any In ) oepeiey oooia space topological a precisely, More A n eget We oeoe e of set open some ∃ M ϕ ( esof sets called hr) h olcino hrsi aldan called is charts of collection The chart). mohmaps. smooth U scle the called is manifold α map a α iswti some within lies C → ϕ , k α V rniinfunctions transition elaayi manifolds analytic real aiod nwihtetasto ucin are functions transition the which in manifold, R scle a called is ) α cover n ∈ ϕ , α i.e. R satplgclsaewihi oal like locally is which space topological a is n : h pc ihoe sets open with space the sahmoopimfor a is ϕ U k R α α n stknt ea ag sncsay pto up necessary, as large as be to taken is ◦ ftefloigaetrue: are following the if → U ϕ . U ϕ chart α α β . α R ∩ − scontinuous, is 1 n U of , : β 4 where ( ϕ , M. U n aeoe esof sets open take and β k h maps the α ( U slreeog o h purpose the for enough large is scle the called is hntetasto functions transition the when α ϕ ∩ α U soet-n n onto and one-to-one is β ϕ M ) V U R ϕ α α α → n − α . , . 1 sa is ◦ ..eeypitof point every i.e. ϕ scniuu,i.e. continuous, is ϕ α domain β ( − U . 1 α smooth , R ∩ hc are which n C U R oopen to k β i.e. , fthe of n ) . , are n 2 2 2 k - c Amitabha Lahiri: Lecture Notes on for Physicists 2011 a nmiuul fafnto ntemnfl ssot sw will we as can • smooth we is manifold then the below. which because on see point, useful function each a is if at manifold unambiguously expansion a say Taylor of a Smoothness have converges. i.e. analytic, real are sta h peei opc pc,adteiaeo opc pc un- space compact a of image the Since and space, compact. between is compact homeomorphism map a continuous is a sphere der the that is ing • analytic). (complex holomorphic be mg ( image ilawy enasot aiodwe emninamanifold. a mention we when manifold smooth a mean always will ( chart sphere suppress will we Often smooth. U between are transformations systems coordinate coordinate and local different defined, be can system nate nifiiesrih ie fcus h osrcinwrsfrany for works construction the set course Of . S straight infinite is an it i.e. but , the chart, in single a projection by graphic sphere charts. two the by cover so to do to possible possible not is It eo ewl n oemrhssfo pnst on sets open from find will We zero. on (1 point this on sets tht h qaoilpae h on hr tht h ln sthe is plane the hits it where point The plane. equatorial the ( hits hemisphere it upper the in is point that n n rt only write and 1 . h esnta ti o osbet oe h peewt igechart single a with sphere the cover to possible not is it that reason The Examples: h yia xml famnfl steshr.Cnie the Consider sphere. the is manifold a of example typical The coordi- local a which on space a is manifold a language, this In o h w hrs ewl osrc hti aldthe called is what construct will we charts, two the For eda tagtln from line straight a draw We { R osdrte‘qaoilpae enda the as defined plane’ ‘equatorial the consider A ie hr ( chart a Given (0 n U x , S ope nltcmanifold analytic complex with x , α n 1 0 2 ϕ , , , , sasbe of subset a as · · · · · · · · · α R C ) n n . x , x , , e ssatwt h ot pole north the with start us Let . hr sas alda called also is chart A 0). n suigtetastosfunctions transitions the assuming and R n n S ϕ ) +1 n 1 ∈ U o hr rudsm on namnfl.We manifold. a in point some around chart a for wt h sa oooy samanifold. a is topology) usual the (with ) in ( ti otcnein oda hsfracircle a for this draw to convenient most is It . α } R x S ϕ , hc ssimply is which , 1 n R n ) 2 2 α R and of o hc h qaoil‘ln’i simply is ‘plane’ equatorial the which for , o egbuho fsm point some of neighbourhood a for ) + n P +1 · · · R n : scle the called is . 1 ( + N x R n n oaypito h pee If sphere. the on point any to +1 x sdfie iial yreplac- by similarly defined is snncmat hr antb a be cannot there non-compact, is oa oriaesystem coordinate local 1 ) R 2 > n (2.1) 1 = coordinates )teln setne till extended is line the 0) hnw goetefirst the ignore we when N x enda the as defined , 1 ,ie the i.e. 0, = ϕ S α of n ◦ P ϕ stereo- oopen to β P, nthe in − 1 the . to 2 2 2 5 c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 • that find we is pole south the from projection stereographic the Similarly, o h ae ewl osdr aiodsc htgopcomposition group ( that map such a manifold as consider) written will we cases the for an is of sphere the functions differentiable are These poles. the is without projections sphere two the the between is function sets transition two the Then the between overlap The write we If that (Exercise!) plane find equatorial we (source on the sphere on of coordinates the similarity the reaching using the Then before through passes through. point) point). first passed (image line has the plane line hemisphere, the equatorial lower which the sphere on the points on For point the of image 6 aaee otnosgroup continuous parameter iino h ieso.Teprmtr r lbl u ontin not do but global, system.) are coordinate global parameters a The form general dimension. the of nition the by parametrized S | nte a fdfiigaLegopi osatwt an with start to is group Lie a defining of way Another dimension z A n | { } \{N 2 i group Lie ≡ ϕ ϕ  N S ϕ 1 sgvnby given is : : S − x S ◦ x 2 x n x ϕ 1 n ftegroup, the of 1 {}→ \{S} n 1 x , x , N dmninldffrnibemanifold. differentiable -dimensional  ,y x, sagroup a is adonly (and z 2 2 2 : , , = R + · · · · · · ) n  7→ · · · \{ x , R x , 1 0 xy n + − x n n → } , +1 2 +1 n − G  x elcniuu variables. continuous real ) 1 n 1   1 hc sas moh(elanalytic (real smooth a also is which , x ssmooth. is R 7→ 7→ − dim = · · · n n G +1 x \{   , 1 hc sagopta a be can that group a is which + 1 z 0 1 1  x } − − 2 x x in n G z , R 2 +1 2 = x Ti sadffrn defi- different a is (This . x x n R 1 1 1 (1 1 ,  , n fteiaeo point a of image the of · · · · · · − \{ hpe .Manifolds 2. Chapter − , 7→ ( 0 , x x , } + 1 1 1 1 . | x x ) ) z hssosthat shows This − z 2 2 n n | +1 2 +1 = x x . 1 1  + 1 1  − n . . scalled is x x 1 1 (2.4) (2.2) (2.6) (2.5) (2.3) n 2 2 - c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 oriae n 3 and coordinates on restriction ( priori of map a sition definition an another without a but is group, as there group also Apparently Lie are are groups a we such manifolds. all if analytic manifolds because , smooth real groups. finite as Lie in them dimensional only define finite interested to about sufficient talking is it are Further, we if objects, same functions the φ of element another be a e aiod o xml,lna obntoso ouin of solutions of man- combinations a form linear region ifold. example, some outside For vanish which Schr¨odinger manifold. a ten an contain not does which identity the subgroup. of a neighbourhood open an find to hc r ohoe n lsdare manifold closed connected and a open of both subsets only are the which Alternatively, sets. open of • manifolds. are spaces) ierflcin oet rnfrain,i -iesoa con- manifold, 6-dimensional 6-dimensional a is a group Lorentz is full The transformations, union manifold. nected disjoint Lorentz the reflection) as time written be manifold, can 3-dimensional it a since also connected SO(3) plus is not rotations is dimensions, of but three group the in O(3), reflections manifold. connected dimensional ( = ( = n hs entoso i ru r qiaet ..dfiethe define i.e. equivalent, are group Lie a of definitions These as written be can group the of element any Then h pc ffntoswt oeseie rprisi of- is properties specified some with functions of space manifold. The 2-dimensional a is M¨obius strip The manifold, smooth of a space phase group The Lie a makes definitions these of Any nnt iesoa etrsae ihfiienr eg Hilbert (e.g. norm finite with spaces vector manifolds. dimensional are Infinite spaces vector dimensional Finite O3,tegopo oain ntredmnin,i 3- a is dimensions, three in rotations of group the SO(3), L dmninlLegopi an is group Lie -dimensional A + ↑ φ a ∪ , 1 1 connected S()weePi reflection. is P where PSO(3) , , h ru fpoe n pc eeto)otohoos(no orthochronous reflection) space (no proper of group the · · · · · · ,y x, a , φ , φ n n ) ) are ) r moh(elaayi)fntosof functions analytic) (real smooth are . 7→ oooia group topological N ic h opsto ftoeeet of elements two of composition the Since aiodcno ewitna h ijitunion disjoint the as written be cannot manifold xy momenta. n − N G, ucin of functions 1 scniuu)i hc ti laspossible always is it which in continuous) is atce sa6 a is particles ecnwrite can we n dmninlmanifold. -dimensional a ∅ and n h aioditself. manifold the and (like N g ( dmninlmnfl,3 manifold, -dimensional a b. ) n g hnfra for Then n prmtrcontinuous -parameter ( nwihtecompo- the which in , b = ) g ( φ a ( ,b a, i group Lie g and ( a )where )) where ) G b . must N 2 2 2 2 2 2 2 2 7 , c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 sacnetdmanifold. connected a is the by orthogonal connected. not 8 hsi an is This h pc fall of space The 3 by represented be can dimensions three in Rotations n 2 matrices dmninlLegopadcnetdmanifold. connected and group Lie -dimensional P = n − × I R . elnnsnua arcsi aldGL( called is matrices non-singular real h pc f3 of space The satisfying R T R × = elotooa matrices orthogonal real 3 I . eeto srepresented is Reflection hpe .Manifolds 2. Chapter × real 3 ,R n, 2 2 2 ).