Notes on Differential Geometry

Domenico Giulini University of Freiburg Department of Physics Hermann-Herder-Strasse 3 D-79104 Freiburg, Germany

May 12, 2003

Abstract These notes present various concepts in differential geometry from the el- egant and unifying point of view of principal bundles and their associated vector bundles.

1 Contents

1 The principal bundle 3 1.1 Sections in P ...... 3 1.2 Fundamental vector ﬁelds and vertical subspaces ...... 4

1.3 Bundle automorphisms and gauge transformations ...... 4

¢¡¤£¢¥§¦©¨ ¥¨

1.4 Lifts from to ...... 5 ¥§¦©¨

1.4.1 Push-forward lift to ...... 6 1.4.2 Lifts by global sections ...... 6 1.5 Connections ...... 7 1.6 Parallel transportation in P ...... 7

2 Associated vector bundles 8 2.1 Deﬁnition ...... 8 2.2 Densitized representations ...... 9 2.3 Gauge transformations ...... 9 2.3.1 ‘Constant’ gauge transformations do not exist ...... 9 2.4 Parallel transportation in E ...... 10 2.5 Sections in E ...... 10

2.6 Derivatives of sections in ...... 11

3 Derivatives 12 3.1 The covariant derivative ...... 12 3.2 The Lie derivative ...... 14

4 Curvature and torsion 17 4.1 Curvature ...... 17 4.2 Torsion ...... 18 4.3 Relating covariant and Lie derivatives ...... 19

5 Building new from old principal bundles 19 5.1 Splicing ...... 19 5.2 Reduction ...... 21 5.3 Extension ...... 22

A Construction of principal bundles 22

Notation: Throughout we consistently use hats and overbars on symbols as follows: hats for ‘raised’ (i.e. lifted) maps of the base or ‘raised’ sections (equivariant functions on total

space), overbars for ‘projected’ maps or local representatives of sections. The differential

of a map is denoted by , its evaluation at by , and its application to a tangent

¥§¦©¨ ¥ ¨ ¥ ¨#"

vector ! by . If is a vector ﬁeld with , this expression is also

¥ ¨ written as % .

1 The principal bundle ' Let & be an -dimensional manifold which we think of as representing space-

time. Most of the differential geometric structures on & can be understood as

originating from the structures encoded in the various principal ﬁber bundles over &

& . On of the most important examples of a principal bundle over is the bundle ,©-.& of linear frames, denoted by ()&+* , where we associate to each point the

set of all bases (here called ‘frames’) of the tangent space /102)&3* . This set is in

bijective correspondence to the group 45(6)7'98%:;* which acts simply transitively on

the set of bases of any ' -dimensional real vector space. The explicit construction of a principal ﬁber bundle in terms of an open cov-

ering of & and transition functions on overlaps (which we call the coordinate construction) is e.g. described in chapter I, section 5 of the book by Kobayashi and Nomizu [5]. It should be familiar. For the convenience of the reader we present in a self contained fashion the details of this construction in the Appendix to these notes. Those not familiar with the notion of principal bundles should deﬁnitely read the Appendix ﬁrst.

Here we only remind that the ﬁber bundle consists of the total space < , with

a free action of a Lie group 4 , such that the quotient with respect to this group = action is given by the base manifold & ; the projection map is denoted by . This structure is pictorially represented by the following arrangements of spaces and maps: _

o ? ?

<< 44 >

& (1) < We write 4 ’s action on on the right (which means that in the coordinate con-

struction of < the transition functions act from the left):

)§EF8HGI*;JBKGML$EON PQR!)¤GI*

43@A

where =TSUQ5RNV= (3) G

The set GZL[4\P¤N^]%G_L`EbacEd-©4fe is called the ﬁber through , and the projection

4 & map =gP

1.1 Sections in P

lVmnBh< =OSUioNqpsrtavu <

A map ijPk& obeying is called a local section of (over

m mKN\&

m ) if is a proper subset. It is called a global section iff . If a section

=9y{z|)}m~* m@4 over mxw^& exists the set is diffeomorphic to the product ; the

diffeomorphism is given by

y{z

)}m~*8 )7,#8Ec*;JBi )7,I* L$E /IMPcm3@o4CB= (4)

This map is said to locally trivialize (in the sense of making it a product) over .

By deﬁnition of a bundle, any Yo lies in an open over which a local sections

g exist. In contrast, global sections exist iff is diffeomorphic to , which is

usually a very strong restriction.

Given two sections and over , they are necessarily related by

{ V {1 {X (5)

for some function d!D . The two local trivializations are then related by

{

#¡c;¢ #% {s¡cX£ (6)

1.2 Fundamental vector ﬁelds and vertical subspaces

The Lie algebra ¤ of induces vector ﬁelds on , the so-called fundamental vector

¨©ª¤

ﬁelds, as follows: Let ¥$¦k§ be the exponential map on and ; then the ¨

fundamental vector ﬁeld, ¨!« , corresponding to , is deﬁned through

¤¬ ¤¬

¯®I°

¨ ;¤ ¨¯%5£

¥$¦k§ (7)

° ±²I³

From (7) it follows that the fundamental vector ﬁelds are ´ -equivariant, with re-

spect to right multiplication, in the following sense:

µ¶|·

{

¤¬ ¤¬

% ^¸s´ ¡ }¨¹ º¡»X£

¨ (8)

·¾½

¤¬

¼ ¨

Clearly, annihilates all « and by a simple counting argument they actually

·¾Ä Ä ·¾½

HÈ ¬

¼ UÅÇÆ ¼ A

span ¿[¥ºÀÂÁ»¥Ã of at each point . We call

½ ½

·¾Ä

¬ «Î¤¬

foÐ1¨_¤2Ñq¿[¥ºÀÂÁ2¥Ã ¼ ´!Ì Í¨ ÏÅ ¤ËÊ (9)

the vertical subspace at ¬ .

ÄÛÚ

µ6×ÙØ

¬ ¤¬

Since, by definition, ÒÔÓÖÕ is just the flow for the vector field

ÓÖÜ

± ¨

« , it is easy to calculate the commutator of two vertical vector ﬁelds by using (8):

ÄÛÚ ·

×Ø

« « ß « «

®I° ®I°

¨ Þ Ò Õ Þ Þ

° °

ÓÖÜ

±²I³ ±²I³

° °

Ý Ý

ß§« à ß

¨cÞ ¥$¦c§ ¨¯% ÞX ´

(10)

±²I³

¨k«

which means that the assignment ¨j¢ is a Lie homomorphism from the Lie algebra of to the Lie algebra of vector ﬁelds on

1.3 Bundle automorphisms and gauge transformations

¤¬

áY¯C á Ö¡c

A bundle automorphism of is a diffeomorphism such that

¤¬

º¤¡ â5ãåä æç»è f

á . Such diffeomorphisms form a subgroup of which we call :

¶ ¶

µ µ

fÏ¤CÍájAâ5ãåä fÐÛá át{éF¡TêfÑ £ æç»è (11)

4 Equation (7) and the deﬁning equation in (11) immediately imply that each fundamental vector ﬁeld is invariant under bundle automorphisms, i.e.

ë[ìHí»î7ïð6î¤ñIò%ò;óôïð6îë9î¤ñIò%ò (12)

By definition, automorphisms map õ fiberwise, i.e., any two points in the fiber

î7ù{ò ùYúoû î7ùFüò ùXüFóþýë î7ùIò ö ÷{ø

ö1÷{ø , , get mapped to another single ﬁber , where and

ëjÿ»ûý û ëYó ë ý

¢¡ ¡ö the diffeomorphism is implicitly determined through ö .

This can be conveniently expressed by the commutativity of the following diagram: ¤

õ õ

¥ ¥

û û

û / £

¤ (13)

ë ë ý ú¨§ © 6îû+ò ë

We say that covers . Alternatively, given ¦ , we say that is a lift

î ò ë gú î ò ë ý Yó

õ õ ¦ ¦

of ¦ in iff and . Occasionally we also write for

ë ë ë

. Clearly ý is uniquely determined by . Moreover, given two automorphisms

ë ë ë ë ó ë ý ëý

ö¡ ¡ ¡ ¡Îö

ø ø ø , their composition clearly satisﬁes , which means

that we have a group homomorphism

ÿ î ò §© 6îû+ò ë! ë#"ý

õ (14)

A gauge transformation is a bundle automorphism ë which covers the identity,

ëAó%$'&)(

i.e., for which ý . We deﬁne

* + î òÏÿ¤ó-,ëjú î ò/. ëAó

ö0¡ ö21

õ õ (15)

which is called the group of gauge transformations (not to be confused with the

î ò

gauge group 3 ). Being the kernel of it is a normal subgroup of . Each

ë ë54^ÿ 3

gauge transformation uniquely determines a function õ through

ë9î¤ñ{òó ÿ2ñ76¯ë84î¤ñIò ë94 î¤ñ76:còfó;: ë84î¤ñIò': * + î ò

÷{ø õ

, satisfying . Hence is isomor-

3 < phic to the group of mappings õ , which are -equivariant in the following

sense:

ë84 ó & ë84

¡>=? ¡ ?A@)B

< (16)

and whose group multiplication is just pointwise multiplication in 3 . We shall

ë84 ëjúC* + î ò

address as the 3 -form of . MONQPI'R¢K

1.4 Lifts from DFEHG I8JLK to

* + î ò3ó STUWVTAX}î ò ©ZY[+\T¯î ò^] § © 6î`_jò

Since we have isomorphic to

î òba*c+ î ò

õ õ

. A priori need not be surjective, which means that there û

possibly exist diffeomorphisms of which do not lift to automorphisms of õ . 6îû3ò

However, all elements in the identity component of § © do have a lift, and are

©dY[+\T¯î ò therefore contained in , as we will show below (c.f. 1.6). But note that

5 rstunwvOp

even if each element of some subgroup egfihkj l mn`oqp lifts to , the lifting

e rstunwvOp

map fyx need (and generally will) not be a group homomorphism.

j l mn`oqp rstunwvOp There are two particular examples for lifting prescriptions x which frequently occur in practice and which we therefore wish to describe sepa- rately and in more detail.

1.4.1 Push-forward lift to zn`oqp zn`o~p

As ﬁrst example consider the case v|{}zn`oqp , where denotes the bun-

^j l mn`oqp ) rstunwzn`oqpbp

dle of linear frames over o . Let , then a lift nAp{ QA

of is given by ‘push-forward’: (cf. formula (70)). This

{} F{ O A8 x

clearly satisﬁes and , where

p0czn ¢¡b£¤p {nH j l m¥n`oqp rstunwzn`oqpbp x for . Moreover, the map so de-

fined is in fact an injective group homomorphism. We may thus regard j l m¥nò~p as ¦ §s2nwznòqpbp subgroup of rstunwznò~pbp which intersects only in the group identity.

This implies that rstunwzn`o~pbp is a semi-direct product of the following form:

rstunwzn`oqpbp¨{©¦ §sinwvOp>ª¨«cjl m/n`oqp (17)

¬ ¬

j l mn`o~p rstun®¦ §s¯nwzn`o~pbpbp5¡ n®°p±n`²p³{©¥´[ n`)pWµ·¶²¸

x (18) ¹0n`o©¡bºp

Note that if we consider a subbundle of zn`o~p , like e.g. the bundle of o

orthonormal frames with respect to some (pseudo-) Riemannian structure º on ,

¹»n`o¡bºp º

then Q no longer acts on unless is an isometry of . Hence only the j l mn`oqp subgroup of isometries of º in can now be (homomorphically) lifted by the ‘push-forward’-prescription.

1.4.2 Lifts by global sections

As second example consider the class of cases where the bundle (1) is trivial. Then

¼©½o v ¾½¿

a global section x exists giving rise to a global diffeomorphism

¿Å

o À v ¾·¿Án Â¡9p{q¼n ÂCpÃA j l mn`oqp ½Ä ÆrstunwvOp x

, . Given , a lift

of is deﬁned by

¿Å

µ·¶ Ä

¾ ´ ¤¾·¿Án Â¡9p>{ n®2n ÂCp5¡9Çp ¿ (19)

or, equivalently,

¿Å

nw¼n ÂCp2Ãp³{%¼n®2n ÂCpbp½Ã¸

(20) ¼¢f

This construction clearly depends on ¼ . To see how, consider another section .

Èo ¼ n ÂCp{%¼n ÂCpÃHÈ¤n ÂCp f

Clearly there is a unique function x such that . An

¿uÉ Å ¿Å

Ä Ä

easy computation shows that the maps and are then related by a gauge

¿ Å ¿Å

½Ä {[>¯Ä

transformation through , whose -form is given by È¤n®²µ·¶n ÂCpbp'Î`µ·¶Ì¡ Êinw¼n ÂCp2ÃÇp³{Ëµ·¶ÌÈ¤n ÂCp±Í (21)

6 which indeed deﬁnes a gauge transformation since ÏWÐ satisﬁes (16). Hence global

sections deﬁne a map Ñ Ò Ó¥Ô`Õ~Ö#×ÙØÚÛuÔwÜOÖ (not a group homomorphism in general!) which is unique up to the obvious freedom of composing each lift with

elements in Ý ÞÚ2ÔwÜOÖ .

1.5 Connections

A connection on Ü is a smooth assignment of a so-called horizontal subspace

ßáà à à

ÔwÜ#Ö ã ä of â for each , which is complementary to , and right-invariant, i.e.

satisfying the two conditions

ßà/è àáé à

ä â ÔwÜ#Öëêãíì7Üïî

Ô å8Öçæ (22)

à àó

ß é%ß ñ½ô Ô å®å8Öëæ;ðñ5ò (23)

Smoothness means that locally the subspaces can always be spanned by õ smooth

vector ﬁelds. Such horizontal subspaces can be uniquely characterized as the anni-

÷ Ü

hilation spaces of a ö -valued 1-form, , on , satisfying

é ù>î

Ô å8Öæø÷¥Ô ùú>Ö (24)

éþý ÿ

¡£¢¥¤

Ô å®å8Öûæüð ÷ Ô Ö÷gô

ñ (25)

à ßáà

Vertical and horizontal projectors into each ä and are then given by

à à à © à

§¦/Ôã°ÖÆæ â ÔwÜOÖ× ä î ¨ × Ô ÷¥Ô ¨ ÖbÖ9úÔã°ÖÁî

Ü (26)

à ßáà à © à à

Ü§ OÔã°ÖÆæ â ÔwÜOÖ× î ¨ ×¨ Ô ÷¥Ô ¨ ÖbÖ Ôã°ÖÁô

ú (27)

¨-ìgâFÔ`Õ~Ö ¨ ÔwÜOÖ

Also, given , there is a unique horizontal lift, ìâ , to each

¢¥¤

ì Ô CÖ

ã , in the sense that

à é

Ô å9ÖÆæ Ô ¨ Ö ¨ î

ò (28)

Ö î ¨

Ô å®å9ÖÆæü÷¥Ô (29)

à àó

Ô å®å®å8Öûæð Ô ¨ Ö ¨ î

ñ ò

ñ (30)

Ô å8Ö Ô å®å8Ö where the right invariance, Ô å®å®å9Ö , is a consequence of and .

1.6 Parallel transportation in P

é é

æ î »× Õ ½Ô Ö ¢Ô!uÖ #" ã ì

Given a curve with , and a point

à à

¢¥¤

Ô CÖ æ$ î ¯× Ü Ô Ö ã . There is a unique horizontal lift, , of with .

Horizontality means that

% ß('*) %

ÿ&%

Ô Ö³ì îÙê ì2 î ô

+-,/.10 (31)

¢¥¤

Ô!uÖFì3 Ô Ö

The end point, " , is referred to as the result of parallelly trans-

ã54

porting ã along . The horizontal lift starting from is given by

àó à

ðñ767 î ñ (32)

7 ;

which is horizontal due to (23). This shows that the parallel-transport map 8:9

<¥=

>@?!ACB ? H@B ? HJI1B

DFE¥G KML

from DFE¥G to is -equivariant.

= =

< < <

> > ?!ACB

A case of special interest is when is closed (a loop), so that ?ONPB and

< < <

> >

?!ACBVU ? B ? H@B ? BSR2T

E¥G

8XWPQ Q are both in D . Then there exists a such that .

The set Y

< <

>[Z >

U]\ ? B_^ U ? Ba`

D 8

Q loop at (33) Rcb is a group, the holonomy group at 8 . From (32) we infer that the elements of

the holonomy group satisfy

< <

>ed >

? BfUhg ? BigMj

E¥G

Q L Q (34)

implying that the holonomy groups at different points in the same ﬁber are conju- Y

gate: Y

>ed >

Uhg gMj E¥G

L (35)

l R

Finally we show how one can use parallel transport to construct a lift k

mMnpo

?ObqB ?wvxB

to any diffeomorphism in the identity component of rSsut . By hy-

R ?wvxB NzjA {}|~

rSsut 9;

pothesis there exists a curve y (a so-called isotopy) such

l l l

U HR2v

that U and . Under this isotopy each point traces a curve

l£ l

? ~B U ? H@B HJIU ? H@B

from H to . Lift this curve to a horizontal curve

? ~B ? H@B ? H@B HRv R R

E¥G E¥G

D 8 D

, starting at 8 . Do this for all and for all .

l l

<¥= <¥= <$=

? B U ?!ACB ? BR ? HJIB ? ~BU ? ~B g

> >ed >

8 8 DFE¥G W k k

Then deﬁne . Clearly and since L

mMnpo

l l l l

?ObqB ? gzBfU ? B g

85W 8 W

k k we have k . Hence is indeed a lift of to .

2 Associated vector bundles

2.1 Deﬁnition

Given a vector space and an action of on via some representation :

T j ?gjpB ?gzB S

; (36)

T b

there is a free right action of on :

T2?Ob B b j ?gjC? jzBB ? gj ?g B zB

E¥G

; 8 9; 85W (37)

We can form the quotient

?Obj j B U?Ob B ¡T¢j

(38)

jP{ gj&{£U j ?gpB &{

8 y 8W y 8

whose elements we denote by y ; clearly, . We shall usually

omit to indicate the dependence of on and , and simply write . The

b v

projection map D of onto now induces a projection map:

Z Z

v¥j j&{ ? j&{ B U ? B

; y 8 9;¦D¤ y 8 D 8 D¤ (39)

8 §

Each point §@¨#©§ª¥«¬ @® deﬁnes a map (also denoted by ):

ª¥«

¬ @®´³ µ¶±¸· §¹³µPº&³

§#¯°±¦© (40)

ª¥«

which is a linear isomorphism between the vector spaces ° and the ﬁber

» §2¨¼© ¬w½x® ¾

of . In this sense the points ª¥« of are ‘frames’ for the vector space

ª¥« ¬ @® © .

2.2 Densitized representations ÀpÁÃÂC¬ ¿p®

Given a representations ¿ , its determinant, , features a one-dimensional rep- ° resentation. Any other representation ¿pÄ on can then be modiﬁed by multiplying

it with a power of ÀpÁÃÂC¬ ¿z® . Representations constructed in this way occur sufﬁ-

ÄÆÅ · À£ÁÃÂC¬ ¿z®iº1ÇÈ¿PÄ É

ciently often to deserve a special name. We call ¿£Ä the -fold

Ä ¿

¿ -densitized representation . A case that frequently occurs in practice is when

¿ ¿Ä Ä

¿Ä is a tensor-representation of and its dual (=inverse adjoint). then describes ÀÊËJ¬Ì°®ÅÍ

the representation of the É -fold densitized tensors. If , sections in the

ÀpÁÃÂC¬ ¿p®iº ¿ associated vector bundle with representation · are called scalar -densities

of weight É .

2.3 Gauge transformations

Î ¯&»±¦»

Any Î@¨#Ï-ÐpÂC¬O¾® deﬁnes a map through §¹³µ&º ®¯ÒÅ·ÓÎ}¬Ò§$®´³µ&º&Ô

Î Ñ¹¬· (41)

§¹³µPº ®ÅÕÎÖÑ¹¬· §×Ø³¿¬Ø£ª¥«® µ&º ®

One easily checks Î!Ñ¹¬· , showing that this is indeed well

±ÙÎ Ï-ÐpÂC¬O¾® » ÎÚ

deﬁned. The association Î¶ deﬁnes an action of on . If is the Û

-form of Î@¨ÝÜSÞ¡Ð¹¬O¾q® , we have

ÎÖÑ£¬· §¹³µPº ®ß¯ÒÅà· §¹³¿¬wÎ ¬Ò§$®® µPº&³

Ú (42) » which deﬁnes an action of ÜSÞ¡Ð¹¬O¾q® , the group of gauge transformations, on .

2.3.1 ‘Constant’ gauge transformations do not exist ÜÞ¡Ð¹¬O¾q®

On ¾ there is not only the action of the group of gauge transformations, ,

Û ±§á×*Ø

but also the action of the gauge group , which we denoted by ¬Ò§¹³!Øp®_¶ . A

¬Ø³C· §¹³µ&º ®¶±

corresponding action of on » does not exist. The obvious deﬁnition

· §(×´Ø³µ&º · §¹³µ&º · §(×Ãâ¥³¿$¬wâ ® µ&º

would only be well deﬁned if we could replace with ª¥«

Û Û

· §¹³¿¬Øp® µ&ºÅä· §¹³¿¬wâpØâ¹ª¥«Ã® µ&ºå$â¢¨

for any âã¨ . This leads to the condition ,

Û Ûìë

æ_çP¬ ®S¯ÒÅéèÃØê¨

which is satisﬁed iff Ø is an element of the normal subgroup

Ûqï ¿

¿$¬wâzØâ¥ª¥«í®#Å¿¬Øp®&å$âî¨ , which we call the center of the representation .

Û Û ® Hence we have an action of æßçP¬ , but generally not of .

Under the name of “constant gauge transformations” it is sometimes suggested Û that a well deﬁned action of on » (as gauge transformations) exists, at least in

9 ñ

the case where ð (and hence ) is trivial. This may sound as if for trivial bundles ôSõ¡öø÷Oðqù

there exists a natural embedding ò¦ó , which is not correct. What is

ó ð

true is that for each global section úûü there exists an injective homo-

¡£¢ ¡¤¢

þ ÿó ÷Oúf÷¦¥@ù¨§ ©zùqû éúf÷¦¥@ù §&þ©

morphism òýóÙôSõ¡öø÷Oðù deﬁned through , ,

¦¥@ù§©zù ©´þ©

or simply ÷Oúf÷ , which is clearly -equivariant. It is also

ÿó

immediate that the map þ is an injective homomorphism. But for a dif-

£ ú§ äûSü ó ò

ferent trivializing section ú , , the embedding would be

¡ ¡

÷Oúf÷¦¥@ù§ ©zù !©"*÷¦7÷¦¥@ùù# þ ÷¦¥@ù$©

, which equals the old followed by a

¡ ¡

ò ÷Oúf÷¦¥@ù¨§ ©pù %© 7÷¦¥@ù)(aþ*+©

gauge transformation whose -form is '& , where

§+(,§ * (#.#*7û . )./

& - - -

& denotes the group commutator . Hence there is no natural ôSõ¡öø÷Oðqù (trivialization independent) way to embed ò into and hence no natural way to speak of ‘constant’ gauge transformations, even if the bundle is trivial.

2.4 Parallel transportation in E ñ

The parallel transportation law for ð induces one for in the following way:

;

0 01 (546*87:9 ÷¦¥@ù 3

Given and as before. Let & ; then we call the curve

; ;

< <

2 = >5? 2 = >5?

0 û (BAC*¥óñD( 0 ÷¦Eùû 0 ÷¦Eù)(54F*

@ &

& (43)

;

(54F* 0 0 ÷HACùI (H0¤1 ÷HACù$*

& 3 & 3 >G?

the parallel transportation of along . Its endpoint, 2 = , is

(54F* 0 3 called the result of parallelly transporting & along .

2.5 Sections in E

;LK ;

;

÷¦¥@ù áûüÙó¦ñ 9 JM ON$QP ñ 9 A map J , such that is called a section in . Since

are vector spaces, such sections always exist, e.g. the “zero-section”, without ñ

RTS JûSðU ó S

being necessarily of the product form ü . A map is called

V @ò

-equivariant iff, W"©X7 ,

KZY[ K

J ÷\© ù J^]U

U (44) V

There is a bijective correspondence (compare (38)) between sections in ñ and - ð

equivariant, S -valued functions on , given by

J£÷¦¥@ùI (_UJ£÷ ù$*

3 3

& (45)

7`9a*÷¦¥@ù ¥

for any 3 and for all .

ð J b J

Given a local section ú in , we can deﬁne a local representative of via

Jc dUJ ú ]

b (46)

e f!gxü For a different choice of local section ú over the same subset , as in (5),

we obtain the relation for each ¥h7if

J ÷¦¥@ù dUJ£÷Oúf÷¦¥@ùj§B7÷¦¥@ùù ÷¦ ÷¦¥@ùù/bJø÷¦¥@ùk] b (47)

m l

As with l , an action of on sections in generally does not exist. Clearly, any l

action on l would deﬁne an action on sections in just by composition.

o p

A useful generalization of n -equivariant, -valued functions on is the notion

o q rtsuqvsuwxzy`{}|~

of n -equivariant, -valued horizontal -forms ( ), whose linear :`¤{o 5n~ space we denote by ¤{o 5n~ . By deﬁnition, iff

{¦~hGaOrZ (48)

OnQ{\C~5 {¦~ (49)

Condition {¦H~ accounts for horizontality, i.e. that these forms are annihilated by

n qr vertical vectors, and {¦$~ is the condition of -equivariance. For we obtain

the space of functions considered above, which we henceforth denote by 6{oI5n~ .

One immediately obtains the useful formula for the vertical Lie derivative in

¨Q{oI5n~ :

£}¤C¥ ¦

GjM X¨n¤{¦©F~5^ ªk©«¬®

G§ (50)

¢¡

2.6 Derivatives of sections in ¯

° l °`± {o 5n~ p

Consider a section in and its associated function on . Let there |

also be given a vector ﬁeld ² on . We wish to deﬁne derivatives of sections

² ²

in l along . This is equivalent to deﬁning derivatives along of functions in

² ²

³{o 5n~ . It is clear that in order to do this we need to lift to a vector ﬁeld

²{B±°k~ on p . Assume some lifting prescription; it must be such that will again be

n -equivariant. That is, we want

± ±

°"~µ{ ¶¸·µ~¨OnQ{\C~ ²´{B±°"~µ{ ¶¤~k ²´{B± (51)

Rearranging the left hand side,

± ± ± ±

½¿¾

,½¿¾

²{B±°Q~µ{ ¶c·,~I °£{± ²¹ º ~I»nQ{\ ~µ{¼ °"~µ{± ²¹ º ~ OnQ{\ ~ °e{¼± ²¹ º ~k

(52)

± ± ±

² ²Àd ² shows that this is true for all ° iff is right invariant: . Hence any

right-invariant lifting prescription will give us a well deﬁned deriv ative operation l

on ³{o 5n~ , and hence on sections in , by deﬁning

ÁÃÂ

¶8 ²{B±°k~µ{ ¶~$Æ ¶t`Ç {¦Ät~k °£{¦Ät~ Å for any (53)

Local representatives can be obtained in the obvious way:

ÂDÈ

²{B±°k~ÉÊ °` (54)

It is instructive to be slightly more explicit at this point. Let Ë be the ﬂow of

Ë ² ² and the ﬂow of (we only need existence for some some ﬁnite interval

around zero.). It satisﬁes

Ë ÌË ÉÇÃ

ÇcÉ (55)

Ï"Ð Î

With respect to the local section Í we can uniquely decompose the lifted ﬂow ,

Ï Ï

Ð#Ñ Ñ¦ÒtÓ5Ó¨Ô Ñ Ð#Ñ¦ÒtÓ5ÓjÕBÖ¨Ñ¦×CØ5ÒtÓkØ

Í Í

Î (56) Ö¨Ñ¦×ÔuÙØ,Õ ÓÔuÚ$Û

with some function Ö , satisfying (constant map onto group iden-

× Ð#Ñ¦ÒtÓ tity). Here we consider ÒÝÜßÞ^à8áâÞ and small enough for to stay inside

Þ . We now have

Û Û

Ï"Ð Ï"Ð

ãÑ Ñ¦ÒtÓ5ÓäÔ Ñ Ñ¦ÒtÓ5Ó Ô Ñ Ñ¦ÒtÓ5Ó Õ,Ö¨Ñ¦×)Ø5ÒtÓ$é

Ð¢æ¤ç Ð¢æ¤çIè

Û6×¤å ÛF×¤å

Í Î Í Í

å å

Ô êëFì Ñ¦ãÑ¦ÒtÓ5Ó8ò Ñ¦ÒtÓjÕBÖ¨Ñ¦×CØ5ÒtÓ$é

çCí îïñð

Ð¢æ¤çIè

Û6×¤å

Í Í

Ô Ñ¦ãÑ¦ÒtÓ5Óeòó6ô Ø

îï Í õ (57)

where Û

óÑ¦ÒtÓö Ô Ö¨Ñ¦×)Ø5ÒtÓÜX÷®ø

Ð¢æ¤ç Û6×

å (58)

å ó Using (57) in (54) and (50) to express the vertical (i.e. ô -) derivative yields

the following general master formula for local representatives of derivatives:

ù úDû ÔvãÑ)û ÓaýÿþQÑ¦óFÓ/û ø

ü ü

ü (59)

óOö Þà ÷ ã The function ¡ implicitly depends on , on the prescription for the

lifting, and on the local section Í . It does not necessarily depend only pointwise on ã but may involve its derivatives, as is does for example in the case of the Lie derivative, where ã is explicitly used to construct the lift (see the section on Lie derivative).

3 Derivatives

3.1 The covariant derivative

According to the foregoing section, the covariant derivative may be characterized

ãÔOã£¢

by putting Î , i.e. by lifting horizontally. We call the corresponding deriva- Dú tive operator ¤ . In order to give an expression that is independent of the vector ﬁeld ã with respect to which we take the derivative, we introduce the exterior covariant derivative, which is just the ordinary exterior derivative followed by pro-

jection into horizontal direction:

Û ö Ô»Û Ô»Û òþ¤Ñ ©¨Ó

ü ü¦¥¨§ ü ü

Î Î Î

Î (60)

where the last equality is most easily veriﬁed by applying both sides to a basis in

Ñ Ó

§ . For horizontal vectors equality obviously holds. For a vertical vector of the

ó ýþQÑ¦óFÓ

ô ü

form the left hand side is zero, whereas the ﬁrst term on the right gives Î

(by (50)), which just cancels the second term on the right hand side. Since the

vectors span , this proves (60). To obtain a local expression we set (61)

which is usually called the ‘gauge potential’. It is a local (on ) representative of the globally deﬁned (on ! ) connection 1-form . As local representatives we now

obtain

$ +*

#¦%'&)( #

# (62)

"-,

. * .£*0*

# ( # %'&/( #21 or ( (63)

Formula (63) should be read as specialization of the ‘master formula’ (59) to the

.45.£6 .7*8:9

( 3 case at hand, i.e. to 3 . This implies so that application of to

(57) gives

<; .7*

( 1

(64)

If we choose a different local section >= , related to the old one as in (5), the

new gauge potential would be related to the old gauge potential by:

?@$ * $

%'AFBED A ( ACBED (65)

where we regarded G to be matrix valued in order simplify notation. A proof for this formula appears at the end of this section

Finally, we remark that the concept of an exterior covariant derivative extends

( &

to all H+I . It then deﬁnes a map

6QP P P 6

*O $ $ +*S $ *CL

( & %R&)( ( &

I INM

KJ H H (66)

where the last equality is again most easily veriﬁed by applying both sides to T

vectors of which either none, one, or more than one is vertical. Clearly, all & -

6 T

equivariant -valued -forms (not necessarily horizontal) are mapped via into

( &

IUM

! H . An important example is the curvature 2-form on which we discuss

below.

.WVYXZ * >\£L

(

[

Let us now prove (65). Let and a curve such that

9*]_^ 9*+.

( (

[ ` [ and . Let further the two local sections and be related as

in (5). Then:

d d

$ $

d d

= =

.7*a b0*0*lkm bo*0*qp b0*0*sr

( ( ( ( ( A+( (

[ [ [

d egf/hji d egf/hCn

$cb $cb

tvuxw .£* ^z*{p ^z* ^z* bo*0*lk

( % ( A8( A ( A+( (

BED

Z¨y

[

d egf/hji

tvuxw .£* ^z* b0*0*s~

( %}| ( A ( A+( (

BED

Z¨y

[

d egf/h

$cb

w w

Z¨y Z8y

uxw

t .£* Z .7*s

( %g w A (

Z¨y

@ Z8y

w w

Z¨y Z¨y

(67)

«v®

£0C@ CE z0¢¡¤£Y¥C¦§@¨ª©¬«8 7¯

Hence,/+ £+/

« µ´<

±°³² (68)

§@¨

z£Y¥

which must be valid for all/¶·R_) FE¢¡, so that we can write, omitting the base {¸ point ,

(69)

¥]º

§@¨

Taking advantage that ¹ can be considered as map into a linear space (matrix alge-

¹ ¹

bra), one may write ¹ for and for . This proves (65).

¥mµ´< ´ °¾½ ¿/

3.2 The Lie derivative E

¿EO¯ ¯ ¿E·Å ¿E·Å

¸x¸x¸

» ¼

Let²/ÇÈµ´Ébe the bundle of linear frames, , over . A point consists

ÁÂ ÂÄÃ ÁÂÆ

of À linear independent vectors (= for abbreviation) in

Å¢8Î ÅC¯

ÊÍËÌ

. As before, ÊË denotes the flow of . We define the lifted flow, , by

¯0Ñ+

ÁÂÄÆ Á¢ÊÍË ÂÄÆ

ÊÍË (70)

¿E·ÅjÔÕ ¿/ Å

ÏjÐ À

which clearly commutes with the right Ï action

¿/·Å ¿7°Ø×

¹ÓÒ7ÁÂÄÆ ÁÂÖ ¹ Æ (71)

A local section assigns a basis ÁÂ¢Æ to each with respect to which we

¿E0] ÇÈ ¿/0C ¿/0 Ç@ ¯

can express the lift in the form (70):

Ç ¿

ÊÍË ÂÄÆ ÊÍË ÊÍË ÂÄÆ ÂÄÖ ÊÍË ÊÍË

Æ (72)

¿E·Å

ÊË ÊEË

where Æ are the components of the Jacobi-matrix for at with respect to Æ

the chosen frames ÁÂ , that is, with respect to . Comparison of (72) with (56)

ÙO¯0¿ECÚ Ç ¿EC Ç

and (58) (in matrix notation) shows that

cÙEÜ

Ö Ö Ö Ö

ÊÍË Û ÊÍË

Æ Æ Æ

and Æ (73)

ËgÝ/Þ

¿E

respectively. The right hand side of the last equation can be explicitly calculated

ß Ê)Ë

Çãåäs ¯ ÇãÚ

using (72). For this we write and note E

©êéK

ªàâá Ü ªà

Üçæè

Ë ÊÍË ÊË Ê Ë

Ê so that (74)

] që·

Inserting this into (72) yields Eoì

ªà

ÊÍË ÂÄÆ ß ÂÄÖ ß Ê Ë

Æ (75)

Ù Ù îí This is just formula (72) but written such that all quantities are evaluated at the

same base point ß . We can now take the derivative with respect to at ,

ð ñ]ò0ó¾ô keeping ï ﬁxed. Recalling that the commutator of two vector ﬁelds is given

ñ+ò0óõôEö±÷ ñÿò

ð (76)

ý ÷cø/ù

ù úsþ

where denotes the ﬂow of ó , we have from (75)

ù úgû/ü

¤ ¤ ¤ ¤ ¤ ¤ ¤

ð¡ £¢ò0óõô ï¦¥]öÚð¡ §¢cò0ó¾ô©¨ ï ¥ ï ¥Cö ¥ ö ï¦¥ ï ¥¨ ï ¥ò

÷ ¢

¢ (77)

¨ ¨ ¨

÷cø

¤ ¤

¥ ¥+ö

where in the last step we used that for a matrix-vùçúgû/ü alued curve with

¤ ¤ ¤

ø ÷

¥ ¥ ¥ ö!

we have . Hence we have shown that

ø ø

úgû/ü úgû/ü

ú ú

¨ ¨#"

öÚð¡ §¢cò0ó¾ô

¢ (78)

)+*-, $/.#01.¦%&#'

Given a coordinate system $£%&(' on with associated frame

öó2&¦.(01.3%4&

and the decomposition ó , one has

¤ ¤

%#¥]ö. ó65 %¥ò

15 (79)

& & which, when used in the ‘master formula’ (59), gives the well known expression

for the Lie derivative in a coordinate frame basis:

>=#¤

798;: : :

< < <

. .Èó2¥

öåó (80) &

We can also derive a useful expression for the Lie derivative of the connection

¤IH

7A@

8CBEDGF

ó òKJ ¥

1-form. Note that for any right-invariant vector ﬁeld ? we have

¤ ¤IH

B DGF

óA¥ òKJ ¥ ó ?

and ? . Then, for any right-invariant ,

¤ ¤ ¤ ¤QPR

7A@ @ @ @

B B B B6STB

8 8ML L 8 8

ö ¥ ö óA¥ ¥ ¥

÷ N ÷ONY÷

P ¤ ¤

ö A¥ ¥

ó (81)

NY÷

The Lie derivative for the curvature then follows from (92):

¤ ¤ P

7W@ 7W@ 7A@ 7V@

B S 8XB 8YB 8 8CB

¥ ö ¥ J

ö (82)

÷ ÷ N ÷

We now pull back (81) using Z , and obtain with (57):

¤ P ¤ ¤ ¤

8 : 7^8T: 7V@

B B B

¥ ¥Cö[ ó2¥ ¥ ¦ö ¨ò

Z (83)

NR\ N]\ N]\

8 7 where is just the ordinary Lie derivative for (vector-valued) forms on , and

where the term involving has been written according to (62) in the last step. This ? expression is valid for any right-invariant ó . Specialization to the case at hand is

now performed by using the Jacobi matrix for , as in (73).

We ﬁnd it instructive to give a second derivation. For this we again write down equation (56), _i`poqij`keOfQf

using the notation _4`badcQegf3hT_ij`keOf¦hml(n :

iqs(t tYu i s i

h>` _ n f o c o (84)

and where vwbxzy{}|p~ . Hence,

~ ~

p £ kC

# g

{ { v

wq w

w (85)

wkxqy wkxqy

~ ~

From (65) we know that

v4 ~ v v {£~ v3

w w w

w (86)

v {Tv

w w

so that the right hand side of (85) is equal to (recall wq )

¢ ( #

~v1w ¡{ ~£ £~ £ £~ v v

w w

w (87)

wkxqy

£G{¥¤v £ ¦§£ where wkxqy is given by the expression (73). The terms involving can be written as according to (62).

What we have said so far is restricted to the case where ¨ is the frame bundle © of © . We made use of the fact that diffeomorphisms of have a natural lift

to ªX«Q©-¬ by push-forward, as expressed in (70). This enabled us to deﬁne Lie

derivatives for sections in all vector bundles associated to ªX«Q©-¬ . But in the general

case no such natural lift exists. As an intermediate example, let © be given a ¯³«Q©-¬ Riemannian metric, and take ¨®°¯±«Q©²¬ , where denotes the bundle of

orthonormal frames. Then the lifting prescription (70), applied to ¯±«Q©²¬ , only makes sense for isometries, so that at this level Lie derivatives can only be taken with respect to Killing vector ﬁelds. This is e.g. the case for spinors, which are

sections in a vector-bundle associated to a double cover of ¯³«Q©-¬ . Generally, you simply cannot take Lie derivatives of spinors with respect to arbitrary vector ﬁelds. Compare [3], where the Lie derivative of spinors with respect to conformal Killing ﬁelds is explained.

If ¨ is trivial, we already discussed in section 1.4.2 how to lift a diffeomor-

phism in a trivialization dependent fashion. Differentiating (20) with respect to the

µ µ¶

ﬂow parameter ´ after replacing with , we obtain for the lifted vector ﬁeld

¸ ¸

« « º6¬ ¬4Å

«¹^« º6¬¼»¾½¿¬§ÀÁÂXÃ§Ä¹ (88) ¹ which is clearly right invariant and tangential to the image of ¹ . When is at the

same time used to derive expressions on © , these become particularly simple due

to the fact that the Æ -term in (57) is now absent. Thus (we continue our notation

ÄÍ Í

ÇÉÈ Ç

¹ ÎË ¹Ê¹ Ï ©

¹ÊÌË , but keep in mind that , i.e. )

· ¸

Ç Ç Ç

«ªVÐ ¬Ò ª Ë «Ë ¬4Å

¹ (89)

Í ÍÓ

«ªVÐ ¬Ò ª Ë

¹ (90) ÕÖ

By construction, these expressions are Ô -, resp. -equivariant, so that any other Ô(«I½4ØÙ¾¬

local representative with respect to ¹×9Î¹>»/½ is then given by applying ,

ÕÖ4«I½ ¬ resp. ØÙ to these expressions.

16 4 Curvature and torsion

4.1 Curvature

The notion of curvature exits for all principal bundles with connection and accord- Û ingly also for bundles associated to them. Let Ú be a principal bundle and the

connection 1-form on Ú ; the curvature 2-form is deﬁned as follows: Ü[ÝÁÞ[ßà

Ûá (91)

ß Ü ßç

â ãXäqåIæ3áKâ where Û is considered as a -equivariant 1-form, and as element in .

In this case it is important to note that, since Û is not horizontal, (66) may not be

applied. Instead, one obtains

ß çì Þ[ß ÜèÞ[ß Þ[ß

â å Û Û Û6éÌê Ûîá Ûðïqá Û6éëê

Û (92)

ä ä3í which differs by a factor of 1/2 from what a naive application of (66) would sug-

gest. The proof runs entirely analogous as for (66). Using a local section ñ and

Þ ì Ü[ÝÁÞ Ü

Ûîá Ûðï Û Û ñó

writing ê we get Cartan’s second structure equation for :

ä¦íQò ò ò ò

ÜMÞ[ß ì

Û Û

Û>é (93)

ò ò ò

Ü çgö

å ôÊáKõ æ ø

acts via the representation ÷ on , where takes its values. To

÷ û}ú1ü ÷(åýû ü

save notation we momentarily write ú instead of , i.e. instead of for

ö ö Þ

æ ü ø þTÿ

û and . With successive application of formula (63), e.g.

ç ç

ù ù

é ÛCå ô ú

ô]å , we can write

ò ò ò

Ü ç Þ ç ç ç ç ç ç ç ç

ù ù

¡ ¡

å ô6áKõ ú å ô å ÛYåõ õTå ÛCå ô ÛCå ô6áKõÉï é ÛCå ô á ÛXåõ ï ú

ò ò ò ò í íò ò ò

Þ ç ç ç ç ç

ù ù ù

¡ ¡

ô å ÛYåõ ú õTå ÛXå ô ú ÛCå ô6áKõ ï ú

ò ò ò í ò

ç ç ç ç ç ç

ù ù ù

é ÛCå ô ú£õ}å ÛCåõ ú¾ô]å é ÛCå ô á Ûgåõ ï4ú

ò ò ò ò íò ò ò

Þ ç ç

ù ù ù

¢¨§

¡ ¡

ô åþ õ}åþ ÿ þ¤£

ÿ¦¥

ò ò ò

ç ç

ù ù

é ÛCå ô ú£þ ÛCåõ ú£þ ÿ

ò ò ò ò

ù ù ù

¢ ¢

¢¨§

¡ ¡

þ ÿgþ þ þ ÿ þ á

£ ©¥

ÿ (94)

ò ò ò

which is a useful equation in applications, in particular if ôÊáKõ refer to coordinate

vector ﬁelds , whose commutators vanish.

ö ç

ã(åýø^á ÷ Applying the exterior covariant derivative twice to a û , we can use

(66) and (92) to obtain

ß ß Þ ÜCçì

à à

÷(å û û (95)

On the other hand, since Ü vanishes on horizontal vectors, it is immediate from the

ß Ü expression (92) that the à -derivative annihilates . Hence one has the so-called

Bianchi-Identity:

ßàXÜMÞ[ßÜ ß çì}ÜMÞ

â å Û é (96)

17 4.2 Torsion

The notion of torsion only exists for principal bundle of linear frames, i.e., if

*)+ -, )/.+

(or subbundles thereof). Recall that a linear frame !"$#&%(' at

6 354

is a linear map from 021 to :

798;:

?> ,B *)+ C>

' '

1=< <

0 3 #A@ @ %A'

4 (97) 7D8

where the map corresponding to the point ! is denoted by . We can now deﬁne a

E6 F

021 -valued 1-form on , , by

8;: 798HG¨IKJ©L M

8ON

F (98) P

The form obviously annihilates vertical vectors. It is also P -equivariant, where

*R2> .TS >

Q 0 0 F 0 P 1 denotes the deﬁning representation of in 1 ; hence .

This is easy too verify:

8 W M M 8

7XGÏ J©L J 7XGÏ JOL GÏ

U U [Z

8 W 8

8 W 8 W

V V

V V V

YP F

F (99)

.\SO]^ >

3 0 P The torsion, 1 , is now deﬁned as the exterior covariant derivative of

F :

*f¦ hg

`_baFY_bFdceP F

3 (100)

ikj l i Fnolm F

For localM expressions on consider and set

7XG¨I

*r¨

3pqlm 3 F m

and . Note that with (98) we get s(tvuxw , which just means that

F F m #&%D' y

the components m of form the dual basis to which deﬁned ; we have

' '

F m %Az pP

z . The local expression of (100) is known as Cartan’s ﬁrst structure

equation:

f g

' ' ' '

3 m {|mF _©mF c F m z

m (101)

>~ >~

' ' '

F } } } _ F } m

Evaluating (101) on vectors and using , m

~ f ~ ~; ~ 2~ | >~

' ' ' ' ' '

} c } {; } } } z

, and m we get (sup-

pressing the index again):

>~ ~$ >~

3 } `{; {2} } m (102)

Finally we remark that F is invariant under the lifted ﬂow (70); this is easy to

prove:

7 JOL M J Mx8

G¨I

8 8

x &

t w t w

t (103)

7XG¨I J M 8

& 4

w t (104)

For the Lie derivatives we thus obtain

> f¦ ¨g

FY` 3`P F

and (105)

18 4.3 Relating covariant and Lie derivatives There is an interesting relation between covariant and Lie derivatives for sections

in bundles associated to o . Recall that both kinds of derivatives were

applications of the master formula (59) which differed in the lifting prescriptions 6¡

and hence led to different expressions for . With respect to a local section

¢¤£ ¢ ¥&¦&§©¨ E6 given by the ﬁeld of bases over , these expressions took the

form:

§ §

ª ª k«¬+®^¯

for covariant derivative (106)

The local expression for the difference between covariant and Lie derivatives of a

section · is then given by

¸;¹ ¹

·«º¯ ·»|¼ ½¯¯ ·

*¬\¿¾±³¦^´µ¬"¶*x¯ (108)

ÀHÁµ*Â2´µÃ¦ 9¯ *¬Äµ ¾

where the argument of ¼ is the matrix in with components

ª ´µ¬"¶

±³¦ . This matrix can be given a neater form by employing (102), which, writ-

¸;¹ ¸ÅÇÆ ¸

§ § §

ª ª

±¯ *¬\¶ ¦ ± ¬È¶ É¡ ¬

ing ¦ and , gives

§ § § §ÍÌ

ª ª ª

Ë±¯ *¬\¶ ¾Y±³¦ ´µ¬"¶ « ¬ *¬È´-¦ (109)

Hence the Lie derivative can be expressed in terms of the covariant derivative and

the torsion:

¸ ¸

¹ ¹ ¹

·¯ ·«\¼5¯ ¬Î¾ÐÏ ·

x¯ (110)

¸ ¸

Ê Ê

§ §

¬ Ï ¥ ¬ ¨ ¥A¬ ¨

where and stand for the component matrices and ® with ® respect to pÑ¥&¦&§(¨ . This expression should be compared to (80) which, for example, shows that for torsion-free connections we may just replace all partial derivatives on the right hand side of (80) with covariant ones.

5 Building new from old principal bundles

5.1 Splicing

KÓ Ô;Ò ÔÓ

Ò Ó Õ

maps Õ and respectively, we consider the following subset of the Cartesian KÓ

product of ¦Ò and : Ì

We denote points in ¦ÒCÖ5KÓ by pairs , but have it implicitly understood that ÒHÖbKÓ

they project to the same point on . It is easy to see that is itself a principal

Ô;ÒÍÛ»Ô×Ó ÕËØÙXÒA´ÙCÓx¦¡Ø`Õ¿Ò&ØÙ¿Ò?Ë`ÕCÓ ØÙ Óx

bundle over with group and projection .

Ò Ó It is called the splicing of bundles and [2]. But it is also a principal bundle

over ßÍà ( ) with group ( ) and projection

ìØí îí ïðÎí ê

à á á ( á ). We have thus a total of ﬁve principal bundles which can be

organized in the following commuting diagram: ‘

ó ó

å é ò

o ò o

ß ß ß ß

à à2ñ á á à EE y EE yy

E ó y ó

ó E y é å EE yy

EE" | yy

ô ô y

(112)

ß©à ßKá

(Technically speaking: ñ can be considered as the pull-back bundle of either

ë ë

ß ß ß ß

Note that all the constructions in the foregoing sections apply to any of these bun-

ß õì ï ß á

dles. The case of interest is where, say, à is the frame bundle , and any

ß ß á other principal bundle. In physics the associated vector bundle sections of à2ñ then correspond to “ ﬁelds carrying different kind of indices”, “space (space-time) indices and internal indices”. Clearly, the process of splicing may be iterated, for

example ß á may itself be spliced. In this case the “internal” indices are again of

“different kind”.

ß ß ß ß

á àñ á

Connections on à and induce a unique connection on : The hor-

ìØí îí ï ö"÷ ø ø ÷ ø ø ìß ß ï

à á àñ á

éÇúüû6ý åµù éþú

izontal subspace at is the unique subspace åÇù ,

ëCÿ¡ ÿ

ø£¢ ø£¢

ê ö¤÷ ø ø ðqö ìß ï ð ã î¦¥ §

å é

ú û6ý

such that ù ( ). The connection 1-form on

ß ß

à2ñ á

is then given by

ë ë

§ð ê § ê §

à©¨ á á

à (113)

ßà ßKá

Right-invariant liftings of vector ﬁelds on to ñ bijectively correspond

ß ß ß ß

á àñ á

to right invariant liftings to à and , and local sections in correspond

ß ß á

bijectively to local sections in à and in the obvious way. Global sections in

ß ß ß ß ß ß ß ß

á à à à2ñ á à á à ñ exist iff they exist for and ; hence is trivial iff and are trivial.

One can now deﬁne all sorts of derivatives of sections in vector bundles as-

ß ß ß ß

á à á

sociated to à¦ñ , by considering right invariant liftings to and and their

ßOà ßKá

combinations to right invariant liftings to ñ . Consider for example the Lie

ß ðõEì ï ß á

derivative for the case where à and a trivial bundle. The vector ﬁeld

ß ß á

on is then lifted to à according to (70) and to by brute force (88) with re-

ß ß ß

à^ñ á

spect to some global section in á . This deﬁnes a right invariant lifting to .

ß ß ß

àñ á

Given a local section à we have a local section ,

ËìØíhàAîíCáAï ðpì¡ ì+ï?î Ëì+ïµï , with respect to which we can write down local rep-

resentatives. The Lie derivative then has coordinate expressions like:

$&%

õ ð

ù ù

ù (114)

Ëî*)Íî,+

where ( are coordinate-indices (with respect to , which we have chosen to

î. take values in coordinate frames), and the indices - are with respect to . Thus, the Lie derivative acts on coordinate indices “as usual” and does not “see” the other indices (“treats them as scalars”). But we emphasize that this latter property is due

20 to choosing the same section / for trivialization and for the deﬁnition of the lifting

of the vector ﬁeld 0 .

5.2 Reduction Reduction and extension are methods to either diminish or enlarge the structure group, keeping the base ﬁxed. In the process of reduction the bundle is reduced to a have a structure group that is a subgroup of the original one. In the process of extension, one enlarges the structure group to an extension (in the technical sense of the word) of the original group, i.e. the new structure group has the old one as a factor group but not necessarily as a subgroup. We will now discuss these cases in

turn, starting with reduction.

1 4#561728 1

Let 132 be a subgroup of and the embedding homomorphism.

: 1;2 :;2 < =;2

A 192 -reduction of is a principal -bundle over , with right action and

? @A5B: 8C: 2

projection 2 , and a map , such that

>LNM'O O

>LSRUTWV

2 2$X 2

@&PQ@ = 1

4FEG5H=JIK (115)

2[Z @&PQ?

4Y4FEG5H? (116)

: : 1 : 2

In other words, 2 is a subbundle of . If the only existing -reduction of is for

: :

132$P\1 , is called irreducible. If is reducible, an interesting question is what

<^]`_ba _ca the irreducible subbundles are. If : is trivial, the subset is a -reduction

of : . We call it the identity reduction. A useful general result is the following:

1 1#db1;2

Let 132 be a closed subgroup of such that is contractible (as topological

1 1 2

space). Then a 2 -reduction always exists. This can be applied to being the

1CPe13f =#E

maximal compact subgroup of the Lie group 1 . The example ,

Dg D

Msr*t

1 Pih E 1;db1 f

2 2kj nbp©q

and Pmlon shows that the frame bundle always

always admits a Riemannian metric.

: 1 2

Clearly the inverse procedure is trivial. Given 2 , we can always embed into

:#2 :

a larger group 1 and regard as a reduction of in the obvious way. We then say

: : that is a prolongation 2 (the word extension is saved for the process described

below). It is important to note that the question of admittance of a reduction is

1x2 2zy{192y|1 y

dependent on the group 1 one starts from. Given ( = subgroup

192 : : :x2

of) and where :#2 is a reduction of ( an extension of ). Then it might

172 2 :#2

happen that : admits a -reduction whereas does not. As explicit example :

we take as 2 the non-trivial double cover of the circle (the edge of the closed

} ~

Mobius¨ strip) regarded as a principal bundle over q . Being non-trivial it does

} ~ not admit an identity-reduction. Embedding in the circle group q , we obtain

it as a subbundle of the 2-torus which represents : . But the torus is the trivial

~ ~ q q -bundle over and clearly admits an identity-reduction.

Since this is an important point, we also want present this rather trivial example in some analytic

detail. The base, , is the circle , is the connected double cover of which is itself a circle,

and is the trivial circle bundle over , which is the 2-torus. We represent all circles by for

& c¦B

6 o

. The generator of the structure group ' of is written as . Then we have

^ b

(117)

¡ ¢¡£B¤¦¥

(118)

¨,©ª

(119)

¬«¡®

§ §

(120)

w´

¶¤

(121)

¶²©

(122)

which satisﬁes (115,115). The identity reduction of to the identity-bundle over (which is a

circle) is then given by

B ¸

F¹Aº (123)

5.3 Extension

¼ ¼ÂÁ ¼

½¿¾À» Ã

Let » be a group and a surjective homomorphism. We call the

¼mÄÅ ¼;Æ ¼ ¼

½ Ã Ç Ç

» » »

kernel of and have » . A -extension of is a principal -bundle

ÉxÊ

È » ÍÎ¾x»Ç Ç

over , with right action and projection » , and a map , such that

ÏÐFÑ Å

¼ÜÛ

É3ÒÔÓ ÉxÊ

ØÚÙ

Ë ×

ËÕÖ Ö

¾ Í Í

» »

» (124)

ÏÐYÐFÑ Å

Ì ÌÞÝ

¾ Í

» (125)

Å Å

¼ ¼ ¼

Ç Ç Èàß Ç Èmß

» »

If is trivial, , a » -extension clearly always exists: just set

Ïá ÑâÅ\Ïá Ï Ñ*Ñ

Û Û

Ø Ø

Í ½ »

and » .

¼ ¼

Ç È

An important example is given for , and »

(a double cover of ). In this case the -extension of È is also

called a spin-structure. If È represents a 4-dimensional space-time one has instead

ÅeäâåìÏ,í Ñ ÅïçðèWéëê$Ï,í Ñ

¼ Û¦î ¼ Û¦î

and » . Since 3-dimensional orientable manifolds

ãAÏ Ñ have always trivial È , they in particular always have a spin-structure. For non-

compact 4-dimensional space-times it is known that they admit a spin structure if

ãAÏ Ñ and only if È is trivial [4].

A Construction of principal bundles

Let È be a manifold (later to be called the “base”) and a collection of open subsets

ñbò6óõô ò¬óûÅ

óýÔþ

Ùùøûú ø

öð÷ È ü È

È which cover , i.e. ; the set is just some

¼ ¼

index set. Let further be some group; we consider the collection of sets ß ,

òó ò£¢¥¤Å§¦

Û*ÿ

ÙQø Ù ø ÷

for all ÷ . For each ordered pair for which we are

¢ ò¬ó ò£¢

Á ¼

¡ ¾

given a function ¨ , such that the following three conditions are ¼

ò¬ó óbó Ï $ÑâÅ

× ©

¨ (126)

¢$Ï ÑÅ ¢ Ï $Ñ ò¬ó ò¢

ó ó

× ¨

¨ (127)

¢$Ï Ñ ¢'Ï ÑâÅ ó Ï $Ñ ò¬ó ò¢ ò

¡ ¡

¨ ¨ ¨ (128)

Next we take the disjoint union of the sets :

)*+

"!$#

(129)

&%(' . and “glue” the different pieces together by using the functions 0/ . The mathematical expression for “gluing” is to identify via some equivalence relation. The

equivalence relation that we use on is deﬁned as follows:

)*+213+54 ) 89+2:;+=< 1 : 4 ) :

! !

,76 ,?> . , and 0/ (130)

That this is indeed an equivalence relation is directly implied by (126-128): Re-

)*9+21£+54 )*+213+54 ,

ﬂexivity, i.e. that ,6 , follows from (126); symmetry, i.e. that

)*+213+54 ) 89+2:;+=< ) 8+2:C+=< )*+213+54

, ,B6 ,

,B6 implies , follows from (127); and ﬁ-

)*9+21£+54 ) 8+2:C+=< ) 89+2:;+=< )ED£+GFH+=I

, ,6 ,

nally transitivity, i.e. that ,6 and imply

)*+213+54 )ED+GFH+=I , ,J6 , follows from (128).

Now we consider the space of equivalence classes, which we call K :

"!LNM 6

K (131)

We call it the principal bundle for the data: O (the base), (the ﬁber or the

PQ P(.

R ;R

group), (the cover), and 0/ (the transition functions). To indicate this

) + + +

, K O PQ P(.

R R

dependence, one could write 0/ .

)*+213+54 )*+213+54

U ,VS K

We shall denote the equivalence class of ,TS by .

KYXZO

There is a natural surjective map W , given by

) )*9+21£+54 1N@

U ,V , W (132)

This is obviously well deﬁned on equivalence classes. The preimage of some point 1

S[O is given by

) 1 )*+213+54

, U ,V

W\] (133)

%0_

`aSb

if . This set is homeomorphic to ; though there is no natural homomor-

`cSd fe

phism, since if / , we could have just as well written down (133) with

8 *

instead of . According to (130) the relation is

)*+213+54 ) 1 4 + ) 89+213+

# #

U ,V , ,V U .

0/ (134)

^ ^

%0_ %Q_

) 1

W ,

which says that our identiﬁcation of \] with cannot distinguish between

) 1 *9+28 1

,2 SB .

ge /

and 0/ for all for which .

The set K carries a natural right action , given by

+j)<+ )*+213+54 )*+213+54n< @

U ,V ,lkXmU ,V

hiK (135)

)*+213+54 ) 89+2:;+=I 1 : 4

! ! ,

This is well deﬁned since for ,a6 we have and

) : I 4n< ) : IH< )*+213+54n< ) 89+2:;+=IH<

. , . , ,o6 , 0/ , and hence also 0/ so that . Note

that this is true because we used left multiplication in our deﬁnition of the “gluing

t uNv

maps” in (130). For the right action with prqs on we shall also write or just prqs

by juxtaposition of the point wq[t with and a dot in-between:

vAx v~}

tYyzta{$w|yzu w9cwp

u (136)

}

uAv w dw wqit p

This action is free on t (i.e. for some implies ) and simply

} }

w£{wCJqc

transitive on each ﬁber 9 (i.e. for any two there is a unique

v~}

u wcw pqs such that ).

Our original building blocks – the sets cs – can now be seen as chart

images of bundle-chart maps:

x } } }

9y sd{ { {5pn|y {5pn

(137)

t Jgs

which are equivariant under the right action of s on and (the latter being

} } }

u-v {=C27|y {=npn

the obvious one: pC{ , which we also denote by ):

{¡ prqs¢

gug¥u (138)

} i£¤£¥H

In the overlap the different chart maps are related by transition

¥ functions, which are clearly just given by the -functions that we used to glue

the different patches:

} } } } }

¥ w2 ¥ wC{¡ ~w?qi g£¥nC¢ w9 (139)

The transition functions on overlapping charts are given by left s -multiplications:

x} }

g ¦g££¥Hs y g£¥n?s

} } }

¥ ~pnC¢ {5pn§|y { (140)

This follows directly from (130). The converse map is given by

x } }

¦g££¥H?y g£?£¥n

} } } ¨ }

(141)

ªC ¦«yzt

A local section of t over is a map , such that ¢

7ª«¥¬® ¯ (142)

¯±°Q²

The chart maps deﬁne obvious local sections ( identity in ):

} x } }

ªC {=( { {=(©¢

(143) ¥

The relation between the sections over and is then given by

} } }

ª³¥ 9ª ¦ ¥ (144)

´µ;¶ ·¸¹zº»¼ ¶ ·3½=¾¸¿¹zº»¼ ºCÂ º ¶ ·£½=¾¸¹

»¼

µÁÀ À

This follows from (141), since µ

»¼

º ¶Ä½2·3½=¾¸Å3¹ ¶ ·£½5Æ½Çº µ3¶ ·¸2¸Å¹ ¶Ä½2·3½=¾(¸Å;Èº µ¶ ·¸9¹´ ¶ ·¸È0º µ¶ ·¸ º

ÂÃ Ã Ã Â

Â Â Â À

µ .

´ Ï ÐYÑ ¶ ¸

ÂÊÉÌËÂ¥Í Î ËÂ

Conversely, given a local section , any »¼ can be

´ ¶ ·¸Æ ·Ó¹hÑ9¶"Ï¸ ÆBÐdÔ ´

ËÂÓÒ Â uniquely written as Â for and . Hence deﬁnes an

obvious bundle-chart map:

º ¶´ ¶ ·¸Æn¸ ¹ ¶ ·£½5Æn¸CÕ

Â É Â (145)

Hence we can give the following deﬁnition of a principal bundle:

Ô Ö

Deﬁnition 1. Let Ö be a manifold and a Lie group. A principal bundle over

Ô ÔØ× ÎÙÍÚÎ

with standard ﬁber consists of a manifold Î and a free right action ,

¶EÆC½Ï¸Û ¶"Ï¸¹cÏ«ÈÆ

ÍzÜÞÝ such that the following conditions hold:

âãÆÐcÔ ÏCá3¹¤ÏrÈÆ

1. Let ÏàßbÏCá iff s.t. be the equivalence relation deﬁned by

ß Ô Ô Ö Îåä

, then is diffeomorphic to Îåä (also denoted by ). The canonical

Ñ Ö

projection map É~ÎYÍ is differentiable. Ö

2. Î is locally trivial in the following sense: there is a covering of by open

Â Ä¥Ðcæ ºÂ Ñ ¶ Â³¸ Â[×TÔ

Ë É Ë ÍçË

sets , , and diffeomorphisms »¼ such that

º ¹èº

ÜÞÝ Â Â ÜÞÝ À

À .

é Â ¹

Remark: Sometimes it is more convenient to use the inverse chart maps É

º »¼ Â .

25 References

[1] Ivan Kolaˇ´r and Peter Michor and Jan Slovak:´ Natural Operations in Differ- ential Geometry (Springer-Verlag, Berlin, 1991)

[2] David Bleecker: Gauge Theory and Variational Principles (Adison-Wesley Publishing Comp.INC., Reading-Massachusetts 1981)

[3] Nick Huggett and Paul Tod: An Introduction to Twistor Theory. London Mathematical Society Student Texts 4., Cambridge University Press (1985)

[4] Robert Geroch: Jour. Math. Phys., 9, 1739. [5] Shoshichi Kobayashi and Katsumi Nomizu: Foundations of Differential Ge- omerty, Vol I. (Interscience Publishers, John Wiley & Sons, New-York, 1963)