c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 h form the function the on imagine acting can we so functions, on side, hand right the On with equation at this Applying pullbacks, and diffeomorphism some Given derivative Lie 11 Chapter hr ehv sdterelation the used have we where ewl pl hst h flow the to this apply will We dt d A t f t dt , t

d t =0 ( − φ φ , t − = = [ = = ( − t ∗ ϕ eget we t v u   ∗ ∗ dt v , u d )( f v φ v ( dt dt t ( v d d ) t ∗ f ( f f ( = φ ) φ A cslnal nvcosand vectors on linearly acts ]( f = )

t ∗ = t ∗ t )) f t f = ϕ =0 v  ) φ )  ehv q 1.)frpushforwards for (10.5) Eq. have we ,

φ − ∗ t f t ϕ φ

=0 A =0 = t t f t φ t t ∗ =0 φ − 37

f t favco field vector a of t v − − t

t − 1 =0
− . dt t = t 1
d =0 1 =
∗ φ . v φ ∗ = + φ v − ∗ hntergthn iei of is side hand right the Then u + t ∗ v ( t ∗ t ( u ( v φ A ( v ϕ f f ( − A φ t ∗ sakn flna operator linear of kind a as f ) ) φ t − ∗ (

t
dt )) Q d f . − ∗  t ( , )) e sdffrnit this differentiate us Let t

. f f ( dt t d t f =0 )

.
t ) φ =0
− ∗ , u

t t ( =0 f endby defined ) v  cslinearly acts

t =0 (11.4) (11.2) (11.3) (11.5) (11.1) c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 So So to • field vector a by functions C on derivation The appro- the called fields, also as vector is written of bracket and Lie module the the So on defined. derivation priately a of properties the • as on this only write compared also be can compared We may be which space. can same vectors they the unlike so points, numbers, different are numerator at the in things The 38 aoiiett,w e that see we identity, Jacobi ne.Frfunctions, For ances. fields. vector where o hneaogteflwof flow the along change not f ∞ invariant. , u £ £ hni ses osethat see to easy is it Then i eiaie r sfli hsc eas hydsrb invari- describe they because physics in useful are derivatives Lie ( M and oti a lob aldthe called be also can this So have to shown be can it and derivative, a of look the has This u u n rte as written and sadrvto ntespace the on derivation a is v ) sadrvto ntemdl fvco ed.As,using Also, fields. vector of module the on derivation a is • f , w and 7→ [ = £ u u ( w , v ( £ f v £ u ) + ( u , v ] ( t lim a edfie iial as similarly defined be can aw → £ fv , £ • w 0 u so £ u u = ) ( = ) . f φ ( = ) f u ( t f ( £ ∗ means 0 = f £ lim = ) £ v u . u £ φ + u u £ £ t u ( sadrvto nteLeagbaof algebra Lie the on derivation a is ( v P t u fg oteflwof flow the So v u ag t → ) f [ = v + − ) ) = ) ( = ) 0 v i derivative Lie C a v , u • φ v w £ + t ∗ P ∞ f £ φ u + f t ( £ [ = ] , w t ∗ − M u £ . f u f v hpe 1 i derivative Lie 11. Chapter f f u v , u = ) • + v ) . . ( g , f £ a u Also, ] + £ u preserves . w otefnto does function the so u f ∀ . g ) f ( u of £ , ∈ i derivative Lie u : f g C ) ihrespect with C ∞ , f , ∞ ( M ( rleaves or M (11.10) (11.11) ) (11.7) (11.6) (11.9) (11.8) . ) → 2 , c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 etrfield vector ϕ preserve which fields vector the So that function a of derivative Lie the £ oi etrfield vector a if So ovnec) the convenience), each at vector cotangent • a selects (smoothly) that point. rule a is 1-form A of space the from map linear smooth on a fields is vector 1-form a Alternatively, a 6, bundle Chap. in mentioned we leaving fields • vector the again Thus £ • functions, → M ∗ u u ω v f w fteeaetovco fields vector two are there If iial,avco edi nain ne diffeomorphism a under invariant is field vector a Similarly, : = ie mohmap smooth a Given As 1-forms. for operations corresponding the define also us Let ehv led enthe seen already have We = 0 = = 0 = v R , v 7→ , n [ and smninderir sn h o of flow the Using earlier. mentioned as ω hc salna a rmtesaeo etrfilsto fields vector of space the from map linear a is which ( v £ £ v v sivratudrteflwof flow the under invariant is ) v , w . f ∈ M = 0 C pullback efidthat find we u eko rmteE.(18,wihdefines which (11.8), Eq. the from know we But w ∞ otesaeo mohfntoson functions smooth of space the to df £ ( £ sivratudrteflw of flows the under invariant is M u ( £ ⇒ + ( u u ϕ £ av u ) T + ∗ ω , , v f ∗ ω ϕ £ w av M ( ) ϕ M = 1 v − ∗ = ) − ] v f ω £ = 1 = ) form £ gradient = u ( sdfie by defined is M → [ v u u f P u φ £ £ ( £ f ω + + − f w u T and [ u t ,v u + ) ( omaLealgebra. Lie a form w P av sascino the of section a is a ∗ v v ϕ nain omaLealgebra. Lie a form invariant 2 M £ = ∗ ] = = = ) f ω v av v syadffoopim for diffeomorphism, a (say £ ) . -omfrafunction a for 1-form v . v 0 = 0 = hc leave which ( u . [ ω f ,v u if ( ) U . , ] . w + ) , u ∀ aω u a efidta a that find we ∈ ( and v M f cotangent R ) . invariant, , , v (11.12) (11.13) (11.16) (11.15) (11.14) (11.17) (11.18) ..if i.e. ϕ f 39 2 if : c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 o oriaetasomtosfo chart a from transformations coordinate For by equation last the both multiply of can sides we invertible, are transformations coordinate Since chart. some in etrfield vector on oacat n raigtecmoet f1frsadvector and 1-forms of components functions, the as treating fields and chart, a to going by one old the index the than – rather here system, ambiguous coordinate somewhat is notation the that Note function the Since components that the noting module. find by can a we is independent, 1-forms of space as written be can which and 40 rm1to 1 from l h components the All 1-form arbitrary an For ftecmoet of components the If ecndfieteLedrvtv fa1fr eycneinl by conveniently very 1-form a of derivative Lie the define can We ω so , n = , v £ ω n h rm culydsigihdtecat rthe or chart, the distinguished actually prime the and u i ω dx ω v i i 0 ( 0 A v i = v , = ) j i 0 v v A i A 0 j j i £ = j i = 0 0 v ω ω u = j ∂x ω ( i v ω ∂x = , v v , j ω i ∂x = ) tflosthat follows it 0 ntenwcataerltdt hs in those to related are chart new the in v ∂x df 0 i , ω j v i i j = A v 0 i v 0 r mohfntos ois so functions, smooth are i j i ∂x − =
j 0 ω ∂ ecnwiei hr n o any for and chart a in write can we 1 A ω , , i = = i ∂x v ∂f n write and − ω , i ⇒ u u 1 i =
j j dx i j ∂x ∂x ∂ω 0 ω ∂ ω i i 0 j j j v i hpe 1 i derivative Lie 11. Chapter ( v . i . i ω 0 v i ω . = ) ω i + 0 i A { A i v i 0 0 x u i − j i
= of 0 i ω j 1 } ω =
i v ∂x i i j oachart a to ∂x ω 0 i ∂x ∂v ω = . 0 j nanwchart new a in j i 0 ω j i ω ∂x ∂x ( . j v i schart- is ) . 0 0 j i ω 0 loruns also I v { (11.23) (11.22) (11.19) (11.20) (11.26) (11.24) (11.21) (11.25) i x . 0 i The 0 } 2 , c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 u ewn odfietig uhthat such things define to want we But sn hsi h oml q 1.9 eget we (11.29) Eq. formula the in this Using the to corresponding 1-form the Similarly, get we derivatives, Lie for formula the into this Putting of components the are These of side hand right the with this write of can side we (11.26), hand Eq. right the Equating as chart a in calculated be (11.26), can Eq. side from equation hand this right of the side and hand left the know already We ie ftecodnt ai etrfilsadbss1frs The 1-forms. basis and the fields to vector corresponding vector basis basis coordinate coordinate the of tives ( o h aeo ovnec,ltu rt onteLederiva- Lie the down write us let convenience, of sake the For £ u ω ( ) v + ) ω dx ( £ i £ £ £ u = v u u ∂x u v ( ω ∂ δ = £ dx ( = ) j i ( i dx v u ( = ( = ∂x i ( = ) ω [ = = = = ∂ = j ) i i i.e. , £ £ £ −   v , u δ = u u u 0 u k i £  £ ω ω ω k ∂u ∂x − u u ∂u u ∂x ) ) ) ∂x ∂v ] j ω i i i j ω ⇒ δ k j ∂x ∂ω v v v ∂x ( ) i k j i k j dx i i i ∂ nagvnchart given a in  ∂x ∂u j + + + v i − j j + ) ∂x + k j ω ω ω = v ∂  k i i i ω j v [  ( ω ∂x ∂u v , u ∂x ∂u j j £ ∂x . dx u ∂u ∂x ( ∂ i = u k £ j j t ai oriaeis coordinate basis -th j i j i v dx  ] ∂x
∂v j i i u δ i ) t oriaeis coordinate -th j v i i j . ∂x j j i = ) . ∂ . − . j δ j i { v . x j i ∂x ∂u } . j i  . (11.30) (11.33) (11.32) (11.29) (11.28) (11.27) (11.31) 41 c Amitabha Lahiri: Lecture Notes on Differential Geometry for Physicists 2011 h aeLinzrl si q 1.7,adtesm ecito in description to same (11.29). the leads Eq. and it in (11.27), that as Eq. mention components in of only as terms but rule detail, Leibniz in same this the discuss not will We 42 forms, hr sas emti ecito fteLedrvtv f1- of derivative Lie the of description geometric a also is There £ u ω | P = lim = t dt → d 0 φ 1 t t ∗ ω h

φ P t ∗ ω . | φ t hpe 1 i derivative Lie 11. Chapter ( P ) − ω P i (11.34)