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Home , Ehresmann connection, Linear connection

DGS

PPR

Lectures on Geometry

Phillip E Parker

notes by J Ryan

Mathematics Department

Wichita State University

Wichita KS

USA

philmathwichitaedu

ryanmathwichitaedu

DRAFT

July

MSC Primary C Secondary C C

Preface

This volume is the sequel to and covers the seminar from It

assumes a certain familiarity with its predecessor esp ecially Chapter The

Tangent Bundle

Now we b egin to lo ok at additional structures that can b e imp osed on a

manifold to provide it with geometry They all involve some sort of connection

or generalized parallel translation so we b egin with the most general kind of

connection as conceived by Ehresmann

A

I wish to thank J Ryan for transcribing my lectures into L T X I also wish

E

to thank J Ryan and S Sahraei for help with indexing and for comments

suggestions and clarications iii

Contents

Preface iii

Standard Symbols vii

Preliminaries

Algebra

Analysis

Topology

Connections

Kinds of connections

Horizontal lifts

and holonomy

Parallelism and geo desics

Automorphisms

Exp onential Maps

Denition and construction

The APS corresp ondence

Jacobi elds

Gconnections

Induced connections

Linear Connections

Automorphisms

Bibliography v

vi Contents

Index

Standard Symbols

Number sets and friend

N natural

Z integer

Q rational

R real

C complex

T torus S

Categories

Bdl bundles

cgH kspaces

Grp groups

Hsdf Hausdor spaces

LAlg Lie algebras

LGrp Lie groups sometimes their germs

Md manifolds

Mo d mo dules

Rng rings

Set sets

Vec nitedimensional vector spaces usually real vii

Preliminaries

We b egin by reviewing some relevant concepts and results from analysis

algebra and top ology

Algebra

Analysis

Topology

In a top ological category a pair of maps f g X Y is said to admit a

homotopy H from f to g if and only if there is a map

H

X I Y x t H x H x t

t

H

with H f and H g Then we write f g or just f g and say that f

and g are homotopic

We can also think of H as either

a parameter family of maps

fH X Y j t g with H f and H g

t

a curve c from f to g in the space C X Y of maps from X to Y

H

c C X Y t H

H t

We call f nul lhomotopic or inessential if it is homotopic to a constant

map Intuitively we may picture H as a continuous deformation of the graph

of f into that of g The next result is obvious

Prop osition Homotopy is an equivalence relation on the set of maps

from X to Y



Chapter Preliminaries

Maps in the same equivalence class of are said to b e homotopic

Denition Two top ological spaces X Y are said to b e of the same

fheqg

homotopy type or homotopy equivalent if and only if there exist continuous

maps f X Y and g Y X with g f and f g Then we write

X Y

X Y and say that f and g are mutual homotopy inverses or inverse up to

homotopy

Similarly to the case for maps is an equivalence relation on any collection

of top ological spaces and one sometimes sp eaks lo osely of spaces in the same

class as b eing homotopic

Denition A space in the homotopy equivalence class determined by

fhptg

a singleton space is called contractible or a homotopy point

Theorem A ber bund le over a contractible base space is topologically

ffbctg

trivial

proof

Even so the bundle might not b e a trivial Gbundle b ecause the Gaction

n

might not b e trivial The on R is one example

Connections

Kinds of connections

In mo dern geometry there are various kinds of connections for a given mani

fold M with a bundle structure over it For example

A general connection on any bre bundle E M is a splitting of TE

into the natural vertical bundle and a horizontal bundle If the

splitting is equivariant for the structure group or more generally some

subgroup G then it denes an Ehresmann Gconnection

A principal connection is an Ehresmann Gconnection on a principal

Gbundle P M G

A on a E M V GLV over M with

mo del b er V is asso ciated to a principal connection on the frame bundle

with group GLV All others are nonlinear among which are

the ane connections with G A It is unfortunate that in the extant

n

literature on nonlinear connections for example all

written well after a nonlinear connection is dened to b e a particular

highly restricted type of connection on TM

A Koszul connection is a linear op erator of the type of a covariant deriva

tive on a vector bundle It gives rise to a linear connection on the vector

bundle

A may b e considered as a version of the general con

cept of a principal connection in which the geometry of the principal

bundle is tied to the geometry of the base manifold Cartan con

nections describ e the geometry of manifolds mo delled on homogeneous

spaces Under certain technical conditions they can b e related to the

remaining types

Chapter Connections

We shall consider all except Cartan connections We are mostly concerned

with nitedimensional real vector bundles E vector spaces V so GLV

GLn R GL with n dim V Moreover our main concern is when E

n

TM so the is LM the bundle of linear frames n dim M

and linear connections are Gconnections for a suitable subgroup G GL

n

All pseudoRiemannian connections are connections of this last type

Since the fundamental work of Ehresmann we have had a consistent

terminology for connections on a manifold M A connection on M is a split

ting TTM V H where V is the natural vertical bundle and H is a

complementary subbundle the horizontal bundle Our main ob jective is to

study smo oth general connections on the tangent bundle TM of a smo oth

paracompact connected manifold M We shall use nonlinear in the original

sense of Ehresmann

We b egin more generally

Denition A general connection on a b er bundle F E M is a

fcong

splitting TE V E H E or just V H for short Since V is natural one

frequently refers to H as the general connection The subbundle H is called

the horizontal bund le We usually omit the general and refer simply to a

connection When E TM it is customary to say the connection is on M

To justify calling H the horizontal bundle one observes that for v E

p

H T M is a isomorphism This follows from V ker

v p

Denition A connection H is at if and only if H is integrable

ffcg

In other words if and only if the horizontal spaces foliate TE Recall that the

vertical spaces always foliate

Denition A connection H on E is trivial if and only if E is a trivial

ftcg

bundle and the horizontal sections are the constant sections

Example The canonical or standard connection on M F is trivial

ftcxg

Example As the tangent bundle to any G is trivial the

ftcxg

canonical or standard connection on G is trivial This gives a canonical at

n

connection on R that has straight lines as geo desics no matter what signature

any co existing inner pro duct might have

ftcxg Ex Trivial connections are at

Kinds of connections

Denition If E is a Gbundle then H is called a Gconnection if

fgcg

and only if H g H A Gconnection on a principal Gbundle is called a

g v v

principal Gconnection

Denition Let E b e a vector bundle with b erdimension k Then a flcg

GL connection on E is called a linear connection

k

Ex In terms of the horizontal bundle H is a linear connection if and

flcg

only if H a H for all v E As is customary here we have written a

av v

as an abbreviation for the induced tangent map of m scalar multiplication

a

by a Hint use Ex in

n

Recall the Grassmannian G R G n of k planes in real nspace

k k

Here we are interested in the splitting TE V H where the b erdimension

of TE is k n and the b erdimension of V is k so that the b erdimension

of H is n Thus we shall consider Grassmannians of the form G k n

n

We wish to determine which elements of G k n are admissible as hor

n

izontal spaces Clearly not all candidates are acceptable such as any element

that is not complementary to V Thus we dene the subspace of horizontal

v

Grassmannians to b e the space of all acceptable candidates

Denition The set of all nplanes in G T E complementary to

n v

fhgg

V is called the horizontal Grassmannian and is denoted by G T E

v H v

G T E

n v

Note that since V is natural it is xed in G k n Thus G T E

H v

k

G k n is an op en submanifold of G k n Indeed V H fg

H n v v

The frame bundle LE of TE over E is a principal GL bundle for the

k n

k n

dening action of GL on R As in Ex this induces the

k n

standard action of GL on G k n and G k n

n

k n k

Denition The asso ciated bundle LE G k n G TE is called

h h

fhgbdlg

the Grassmann hplane bund le over E and G TE LE G k n is the

H H

horizontal Grassmann bund le over E

Observe that a connection H on E may b e regarded as a section of G TE

H

over E so as H G TE

H

We shall now determine the subgroup A GL that xes the vertical

H

k n

k

space R and acts transitively on horizontal spaces First we may apply

any automorphisms of V and H separately Second we may add vertical

Chapter Connections

comp onents to horizontal vectors to obtain a new horizontal space In blo ck

matrix form we write

I GL

k

M I GL

n

nk

Thus A is a semidirect pro duct entirely analogous to the usual ane group

H

n

A R n GL

n n

Ex The preceding action is transitive and the righthand factors com

prise the isotropy subgroup A GL GL of any xed horizontal space

o n

k

n

such as R

Therefore the mo del b er G k n is the homogeneous space GL A

H o

k n

The induced op eration on representative elements is given by

I I I

A I B I A B I

Thus G TE is an ane bundle over E Hence a connection H provides a

H

distinguished p oint in each b er a vector bundle structure on G TE over

H

E From Ex we know that ane bundles over manifolds have ane

Frechet nuclear section spaces

An alternative view of a connection is based on the horizontal pro jection

fhpaxg H TE H Axioms for H are

C H EndTE over E so H is a tensor

C H H

C ker H V

Then H im H is the horizontal bundle

Ex H is the horizontal pro jection of a linear connection H if and

only if H a H

av v

Horizontal lifts

Let I M b e a path in M and assume that I with p and

q

Horizontal lifts

Denition We say that has horizontal lifts if and only if for every

fhlg

v E there is a unique path I E such that v E and

p q

is horizontal along

If has horizontal lifts then each satises

What is wanted is an analogue of parallel translation of vectors along

a path In particular we want via pullback if is not

injective providing a dieomorphism P E E Equivalently the set of

p q

all horizontal lifts of smo othly foliates E over I

Axioms Here faxfbg

axioms for a ber bund le

Denition When such a P exists for every it is called parallel

fplltg

transport along

Remark One might consider weakening parallel transp ort to merely

require that P is smo oth a submanifold an embedding etc However this

would severely ero de almost all subsequent results

Denition An is a general connection with

fecg

horizontal lifts of every path and parallel transp ort along every path

Having this horizontal path lifting HPL prop erty is denitely nontrivial

Ehresmann recognized as much by including HPL in his denition of a connec

tion in Which general connections have HPL is neither well understo o d

nor well studied After over years a complete answer is known only for

general connections on TM and that is recent

Let E b e a vector bundle with k dimensional b ers over an ndimensional

k n

When manifold M A typical b er of TE over E lo oks like a copy of R

n

E TM then k n and T TM R for all v TM

v

k n

Now supp ose that we endow R with a p ositivedenite inner pro duct

k n

h i For u v R dene the angle between u and v by

jhu v ij

fanfng cos

kuk kv k

with kuk hu ui as usual so that This angle is the standard

one b etween the vector subspaces lines spanned by u and v in linear algebra

Euclidean geometry which is precisely what we want

Chapter Connections

k n

Consider a k plane V and an nplane H of R that are complementary

k n

V H R Then there are exactly minfk ng nonzero angles b etween

these subspaces

Denition Let u V and let u H b e its orthogonal pro jection

fwongg

onto H The angle b etween u and u dened by is a smo oth real

valued function of u Letting u vary over V the nonzero critical or stationary

values of are called the Wong angles b etween V and H

Ex Starting with v H and v V yields the same set of nonzero

critical values for Thus the Wong angles are well dened for a given p ositive

denite inner pro duct

We shall apply this to general connections for b er bundles E over M

where V will b e the vertical and H the horizontal subspace of a b er of TE

over E The sp ecic values of the Wong angles are equivalently the sp ecic

choice of a p ositivedenite inner pro duct is not invariantly well dened But

whether they are b ounded away from and whether they are constant along

a b er of E is invariant and that is all we shall need

Denition An innerproduct bund le is a vector bundle E over M with

fipbg

a smo othly varying inner pro duct on each b er E More precisely

p p

E E is nondegenerate and symmetric on each b er of E When each

is p ositive denite E is commonly called a Riemannian bundle

p

When E TM then is a tensor on M

Ex Show that the signature of is constant along connected comp o

p

nents of M

We say that a b erwise p ositivedenite is a Riemannian inner pro duct

and one of signature n or n is Lorentzian When E TM the

same terminology is applied to M It is customary to denote u v hu v i

p p

or even just hu v i with p implicit from context one hop es This last notation

extends readily to h i FM for sections of E over M

Example The canonical or standard pseudoRiemannian structure on

fcipg

k

M R is the constant symmetric tensor dened by the canonical or standard

inner pro duct diag of signature p q with p q k on

k

R

Horizontal lifts

Example Let G b e a Lie group with g Any inner pro duct

fciplgg

on g extends to one on G as leftinvariant resp ectively rightinvariant For

example if V W RG we have hV W i hV W i for all g G Such a

g g RG

structure that is simultaneously left and rightinvariant is called biinvariant

n

The canonical structure on R from the preceding example is biinvariant

Thus to assign Wong angles to a connection on E we must make TE into

a Riemannian bundle by choosing a Riemannian metric tensor on E the base

space of TE over E The next result assures us that this is always p ossible

Theorem Any vector bund le E over a smooth manifold M admits a

frbg

Riemannian inner product

Pro of This follows from a standard partitionofunity construction



Theorem A connection H on a vector bund le E is linear if and only

fvcwag

if the Wong angles are constant along the bers of E or vertically constant

Pro of Scalar multiplication preserves angles



Ex Note that having vertically constant Wong angles is also well de

ned for a b er bundle E What can you conclude ab out such a connection

Denition A connection is uniformly vertically bounded UVB if

fuvbg

and only if H is b ounded away from V in each T E uniformly along the

v v v

b ers of E

To my knowledge the next theorem is the rst complete characterization

of necessary and sucient condition for the horizontal path lifting prop erty

HPL in TM as rst given by Ehresmann p

Theorem Let H be a general connection on M and let I M be

fhpltg

a path with p and q For every v T M there exists a unique

p

such that v and T M if and only if H is UVB horizontal lift

q

Pro of Consider the TM over I Note that TM

n n

I R The lab el will b e used to distinguish copies of R The pullback

T

n n n

R R TTM R The general connection pulls back to the family

B V

n n

of horizontal nplanes H for each u R We may assume that R is the

u

T V

n

vertical space and shall refer to R as the basal space

B

n n

We seek a curve c I R such that c v R and c t H

ct

T T

n n

It is convenient to identify R and R Let D denote the usual derivative in

T B

Chapter Connections

n n

R so that D c is the Jacobian matrix of c Then with ct R we have

B

c t ct D ct ie ct is the basal comp onent and D ct is the vertical

comp onent of c t

Now the UVB condition implies the existence of an upp er b ound on kD ck

which is uniform along the b ers of TM With I compact the b ound

may also b e taken to b e uniform along I Applying an MVT p this

implies that kck is b ounded on the part of I where it exists The FEUT

pp f provides the existence of c and the Extension Theorem

p f shows that c extends to all of I

So we may regard c TM over I By denition c is a horizontal

of c is a horizontal section of TM section It follows that the pushforth

v and T M Uniqueness of follows immediately along with

q

from that of c

The converse follows similarly noting that kck b ounded implies kD ck is

also b ounded since c is smo oth



Corollary Each path in M from p to q denes a dieomorphism

fplltcg

P T M T M that is called parallel transp ort along Note that

p q

fhlpeg

as for vector elds If is not injective however this has to be interpreted

via the pul lback bund le TM

Pro of Clearly uniqueness and smo oth dep endence of integral curves on

initial conditions p imply that the set of all horizontal lifts of denes

such a dieomorphism indeed as I is contractible they induce a horizontal

of TM



It is now routine to obtain all the usual prop erties of parallel transp ort as

in for example except of course for linearity of the maps P

Corollary This P in TM along I M with

fplltcg

p and q has the fol lowing properties

Existence and uniqueness for each v T M and each smooth with

p

v there exists a unique smooth curve P v from v T M to

p

T M Alternatively one may regard P v as a vector eld along

q

Invertibility al lowing v to vary over T M the resulting map P

p

T M T M is a dieomorphism Its inverse is parallel transport along

p q

the reverse t t

Holonomy

Parametrization independence if is a reparametrization of then

P P T M T M

p q

Smooth dependence on initial conditions

Initial uniqueness if and are two curves starting at p with

v then P v P v have the same tangent vector at v



Most of these have b een proved already we leave the rest as an exercise for

the reader

Prop erties and give a concrete expression to the idea that P is closely

akin to a lo cal ow This was already seen in the fact that the horizontal

lifts foliate as noted after Denition and in the pro of of Corollary

Let us note that Poor Ch shows that these ve prop erties suitably

mo died may b e taken as axioms for a system of parallel transp orts in a

vector bundle that is equivalent to a connection in that vector bundle Adding

linearity to the second prop erty or axiom is equivalent to the connection

b eing linear

Parallel transp ort had b een generalized previously in at least two main

ways One lo oked at general transp ort along paths and another at parallel

transp ort along general submanifolds

Holonomy

In this section we assume that M is connected and H is an Ehresmann

connection

!

Let P M denote the reduced oriented path space of M An element

of P M is a member of an equivalence class in

f I M j I a closed intervalg

of the equivalence relation E generated by all orientationpreserving reparam

! !

eterizations Supp ose that is dened on I a b and on I c d

! !

with b c Reparameterizing so that b c we dene the pro duct

! ! ! ! !

by concatenation ie the pro duct path rst follows then

!

follows We now drop the arrow and write simply to denote an element

of P M

This pro duct is an asso ciative binary op eration Ex verify thus making

P M a semigroup with each element invertible the inverse of is the reverse

path dened in item of Axioms Moreover there is one identity

element for each p M Thus P M is a groupoid a category in which all

Chapter Connections

arrows are invertible The ob jects are the p oints in M and the arrows are

the equivalence classes of oriented paths

Example A group is a group oid with a single ob ject

Denition The group oid G is called path connected if and only if

fcnctgpdg

each pair of ob jects has an arrow b etween them

Example A transitive determines a connected groupoid

Denition For each ob ject x of G the subgroup oid G x consisting

fvxgrpg

of all arrows x x is called the vertex group or loop group of G at x

Ex Verify that G x is indeed a group

Example The isotropy group of any transitive action is a vertex group

Theorem If G is connected then al l vertex groups are conjugate hence

fcnctcjtg

isomorphic

G y The induced Pro of If G x and x y then

mapping G x G y is easily seen to b e a group morphism Moreover

it is invertible



For a manifold M one may dene the lo op group M p at p M

similarly as the subgroup oid of P M consisting of oriented paths p p

Observe that parallel transp ort denes a group morphism P M p

p

Di E P where Di E is the group of dieomorphisms E E

p p p p

Equivalently dene P P P

2 1 2 1

Denition The image of P is called the holonomy group of H at the

p

fhlgpg

point p M and is denoted by  p

Now apply the preceding theorem Let F denote the mo del b er of E

Corollary If M is connected al l of these groups are isomorphic and

fglhlgpg

the global holonomy group  Di F of H is wel l dened



Note that  need not b e a closed subgroup indeed this already happ ens for

linear connections

In Theorem showed that F is uniformly dense in C M from

which it followed that C M N is uniformly dense in C M N for all man

ifolds M and N Sp ecializing

Holonomy

Lemma Each homotopy class of paths or loops contains a smooth path

fsmhoclg

or loop respectively

Pro of For lo ops based at p M C S M is dense in C S M Thus

each class in M p contains smo oth lo ops A similar argument works for

paths



Denition The restricted holonomy group at p M is the subgroup

frhlgpg

 p  p of parallel transp orts around nullhomotopic smo oth lo ops

o

based at p

Prop osition For the relative topology from Di E the restricted

p

fhmg

holonomy group  p is a pathconnected normal subgroup of  p in fact

o

it is the identity component and there exists an epimorphism M p

 p p called the holonomy homomorphism

o

Pro of Letting p denote the constant lo op a nullhomotopy H from to p

is a path in  p from P to For any h  p h H h is a path from

o

h P h to Thus hP h  p whence it is normal The morphism

o

required for the last claim is M p  p p P

o



Recall that H is the trivial connection if and only if E is a trivial bundle

and the horizontal sections are the constant sections

Prop osition A connection is trivial if and only if its holonomy group

fthgg

is trivial

Pro of Fix p M and dene a map E E v P v where

p

v and p Note that is well dened and smo oth Then

is a b er bundle morphism E M E moreover it is parallel

p

Conversely every parallel eld along can b e written as for some

parallel section E Then parallel transp ort of around a lo op

yields



Prop osition A at connection is locally trivial

ffcltg

Pro of Let N b e a maximal connected integral submanifold of H in E

The restriction j is surjective Indeed given v N and p M let b e a

N

smo oth path from v to p Since P v is horizontal then it lies in N and

P v p Applying the InFT Thm the smo oth submersion

j is a lo cal dieomorphism hence a covering map N

Chapter Connections

Let U M b e connected and op en and take E U E j Now

U U

E is an op en subbundle of E and E N is a disjoint union copro duct

U U

of op en connected submanifolds of N Since maps each comp onent N of

c

E N dieomorphically to U then j is a section of E Moreover it is

U U

N

c

a parallel section b ecause N is an integral manifold of H j There b eing

c E

U

such a parallel section through each p oint of E it follows that H j is the

U E

U

trivial connection



Theorem A at connection has trivial restricted holonomy groups so

ffchg

the holonomy morphism becomes M p  p

Pro of Let I M b e a smo oth nullhomotopic lo op at p Then there

exists a smo oth homotopy H I I M with xed endp oints b etween

H p and H Each intermediate lo op H s is smo oth

s

Given v E let N b e the maximal integral manifold of H through v Then

p

for all s the horizontal lift lies in N Since N is a covering space of M

s

by Prop osition the Covering Homotopy Theorem implies that the

map s t P v t is a homotopy in N that covers H and thus has xed

s

endp oints b etween P v and the constant path v But P v v so parallel

transp ort around is trivial and  p By Prop osition there is

o

a group morphism M p  p p  p

o



f f

Theorem If M is connected with universal covering M and M

fncfcg

M is the covering projection then H is at on M if and only if E is a

f

f

trivial bund le over M and H H is the trivial connection

f

Pro of If H is at then so is H From Theorem the holonomy

f

f f f

group  p of H at p M is the image of M p hence trivial Using

f

Prop osition E is trivial and H is the trivial connection

f

Conversely given v E let N b e an integral manifold of H through

v E Then N is an integral manifold of H through v E whence

H is involutive Now apply the Frobenius Theorem



Consider lo cally convex Lie groups as in Ch One having an

exp onential map exp g G that is a lo cal dieomorphism near is

called locally exponential If such a Lie group acts on a smo oth manifold

then we can dene fundamental vector elds as in Example

In particular  p acts on E Assuming each  p is lo cally exp o

p

nential we can use these fundamental vector elds to build holonomy

algebras and a holonomy vector bundle as in Poor Together with

with a curvature tensor and op erator this should allow us to obtain an

Covariant derivative

extended AmbroseSinger Theorem relating holonomy and curvature as

in Reckziegel and Wilhelmus

Covariant derivative

In this section E is a vector bundle over M

Recall from that the bundle morphism K V E resp ects the

T

bundle structure on TE over E but not the structure on TE over TM

Furthermore K is a b erisomorphism and a version of canonical parallel

translation on a vector space

Denition Given a connection H the asso ciated connector is the

fcnctrg

map

TE E z Kz H z

v

for all z T E

v

Note that H z is R linear in z but is only smo oth in v

v

Prop osition The connector is a vector bund le morphism that re

fcvbmg

spects but not necessarily It respects if and only if H is linear

T

Pro of The rst part holds for b ecause it holds for K

nish as in Poor p f



When E TM a connector may have additional symmetry Recall that

I is the natural involution of TTM given in induced lo cal co ordinates by

x y X Y x X y Y

Denition A connector TTM TM is symmetric if and only if

fscg

I

Ex A symmetric connector is only p ossible if the connection H is

linear

If the connector is symmetric one also says that the linear connection on M

is symmetric

Before stating the main denition we need a common attributive in dif

ferential geometry An op erator is tensorial if and only if it is Flinear

Chapter Connections

Denition Let V X and E The covariant derivative asso ci

fcdg

ated to a connection H is given by

r V K V H V

V

It is tensorial in V but only smo oth in

Ex Show that tensorial in V means that r is well dened for v TM

v

Notice that the right hand side of this formula could b e rewritten as

K H V

TE

where H is the pro jection onto V parallel to H

TE

Ex Write down the sp ecial case when E TM replacing by W

Remark The fact that r is in general nonlinear in is a consequence

of the complete lack of resp ect for by H H and

Example Any connection satises r For a linear connection

fcdxg

r a Ka V H a V a K V H V

V a

In particular r for all linear connections in fact all linear connections

V

share the same horizontal space along the section of E namely the subspaces

of b ers of TE that are tangent to the section Equivalently the horizontal

space along the section of a linear connection is the image of TM TE

Any connection satisfying r for all V X is said to b e preserving

V

These dier the least from linear connections By contrast those with r

V

for even one V X are strongly nonlinear

The next result is clear from the denitions

Prop osition Let E be a vector bund le over a connected M A con

fprg

nection is preserving if and only if its holonomy group is a subgroup of

k k

Di R the isotropy group at R



Theorem There is a bijection between the set of connections H and

fccdg

the set of covariant derivatives r

Covariant derivative

Pro of It suces to show that H can b e reconstructed from r For all

u E dene

p

H f v J r j E u v T M g

u u v p p

This gives a subbundle H TE complementary to V That H is smo oth

u

H H follows from this and is a straightforward verication Finally that

previous constructions

expand this cf Poor p proof of



Thus as usual we shall refer indierently to H or r as the connection

We introduce general connection coecients by

k

i k

V KH V

i

k i k

making manifest the tensoriality in V Noting that K V V we

i

obtain

i k i k k

V r V

V i

i

as the k th comp onent of the covariant derivative

For the record we note that Bucataru and Miron dened parallel trans

p ort to b e linear isomorphisms They derived from that a completely dierent

kind of nonlinear covariant derivative op erator and thence a certain type of

general connection

As the next thread we shall consider pullbacks of connections

Let f N M b e a smo oth map of smo oth manifolds and E a bundle

E over N The over M Recall that one may form the pullback bundle f

space of all lifts of f to E is denoted by E f f E where f E is the

f

space of sections of f E over N and f f E E is the pushforth of f The

following diagram illustrates this situation

f

f E

E



f

M N f

Chapter Connections

Prop osition For any ber bund le E over M and smooth map f

fpbsg

N M f provides a natural bijection f E E If E is a vector

f

bund le then f is an FN module isomorphism

Pro of Let f E For all p N is an element of the b er f E

p p

Thus we may write v E The smo oth map f s has the

p

f p

prop erty that s f and the following diagram commutes

f

f E

E

O

s

M N

f

pulls s E back to f s f E for all p N Thus s E Now f

f f

f s v When E is a vector bundle linearity is obvious

p p



Denition Let f N M b e a smo oth map and U XN Dene

fcastg

f U f U N TM

~

f



TN TM

O

y

y

y

y

y

y

U

y

y

f U

y

~

y

y

y

N M

f

Example Let c I M b e a smo oth curve Then c c D c D

~

fcdotg

where D ddt is the standard constant vector eld on I and c c D

~

Example Left and rightinvariant vector elds on a Lie group G are

flrivfg

sections of TG along L resp ectively R Recall that leftinvariant vector

g g

elds are L related to themselves and similarly for rightinvariant vector

g

elds In general related ob jects are also examples along

Example Supp ose G is a Lie group H a closed subgroup and

fhfvfg

G GH the natural pro jection to the homogeneous space GH For exp

LG G the Lie exp onential map dene a map LG XGH by

d

exptV g V

g

dt

t

Covariant derivative

for V LG and g G ie V with t exptV g GH

g

For each V LG dene V RG by V R V Then for all g G

g g

d

R exptV V V R V

g g g

g

dt

t

Thus V V and V on G is related to V on GH It follows that

g V Ad V g for the standard left action of G on GH Moreover

g

the map V V is a morphism of Lie algebras RG XGH

Example More generally if a Lie group G acts on M then there is a

ffvfg

map g T G XM dened by

d

v exptv

dt

t

op

which is a Lie algebra morphism g RG XM The image is commonly

called the set of fundamental vector elds of or along the action

Ex If the action is eective then the morphism is injective This is

flftg

Lies First Theorem

Note that the action of a holonomy group  p on a b er E is eective

p

by denition Therefore the map p XE is always injective when

p

it exists Thus we may identify p XE whenever it is convenient

p

This yields a smo oth vector bundle over M with b ers p called the

holonomy bund le

Theorem Let E be a vector bund le over M and H or r a connection fpbcdg

in E For al l smooth f N M V XN and E

f

f r V

V ~

denes the pullback connection along f

Pro of First note that as f is smo oth comp ositions of f and of f with

smo oth maps are smo oth The connector here is the same one used for r

so f r is a smo oth vectorbundle morphism is tensorial in vector elds on

N and is smo oth in sections of E along f Therefore it is an op erator of the

same type as r is



We may think of this as describing the covariant derivative of sections along

f with resp ect to vector elds along f and may also denote it by r

f V ~

Chapter Connections

Corollary If g L N and f N M are smooth then

fchaing

g f r g f r

V g V

~

for V XL the chain rule

The pro of is left as an exercise Hint expand b oth sides

Ex There also exists a pul lback connection f r in f E over N Write

down the details

For the next result we rst need a lift provided by the connection

Denition Let V X and H b e a connection in E The horizontal

fhlvfg

h

lift of V is V V dened as the unique horizontal vector eld on E that

pro jects to V Alternatively V XE is the unique vector eld on E such

that V and V are related V V

When E TM the horizontal lift is also given by

w V w

H

v

where w T M and v V p

p

Ex Let U V X and f F Horizontal lifts have these prop erties

fhlvfpg

U V U V

v

f V f V

h

U V U V H

v

Here f f is the vertical lift of f and H is the horizontal pro jection

Ex Verify that to determine if H is involutive it suces to consider

fohlig

only horizontal lifts of vector elds on M

Denition The curvature operator of a connection is

fcog

R U V w V U w

for all U V X and all w E

Note that the curvature op erator is tensorial in U and V but merely smo oth

in w

Curvature and holonomy

Ex For U V X and E we get the usual formula

R U V r r r r r

U V V U

UV

Since H is integrable if and only if it is involutive it follows that R if

and only if H is integrable

Finally let H and H b e two connections with corresp onding covariant

derivatives r and r

Denition The dierence operator of the two connections is D

fdog

r r

We regard D as having two arguments DU r r It is tensorial

U U

in U but merely smo oth in

Denition The covariant dierential is r U r Note that

U

fcdfg

r is tensorial in U

H fz J D z v j z H g Lemma For al l v E

v v v

fadolg

Pro of Let v E z H and E such that r p and p v

p v

If u z T M then z u H Calculating

p v

u r r Du

u u

Du J Du

v

so z J Du v whence z JDu v ker H As maps b oth

v v

H and H isomorphically to T M and z z J Du v this yields

v v p v

all of H

v



Curvature and holonomy

In this section we return to E a general b er bundle

Recall that the connector map on a vector bundle is given by K V

On a b er bundle E M the map K do es not exist we only have the

vertical and horizontal pro jections V and H

Denition The curvature form on E is dened by

fcfg

U V V H V H U

Chapter Connections

It is tensorial and alternating in U and V but merely smo oth on E The

curvature form on TM TM is

h h

R u v u v

w

where w p Again it is tensorial in u v but merely smo oth in w

Some other notes

Notice that R u v is an element of XE which is a Frechet nuclear

p

space Thus there is a welldened notion of smo othness TM

TM XE

p

If E F are Frechet nuclear spaces and f E F then D f

x

LE F may b e dened via the usual limit

Thomas states that we may replace Frechet by Kelleyed

quasicomplete and the ab ove remark will still hold Further

more complete may suce for us

1

Keller says that there is a unique notion of C in these

HLCTVS

Parallelism and geo desics

Now E is again a vector bundle over M

Observe that r if and only if U is a section of H over E

U

Denition We say that is parallel along U more precisely along

fpvfg

the integral curves of U if and only if r Taking U c we get parallel

U

along c When r for all U equivalently when r we say that

U

is simply parallel or for emphasis absolutely or globally parallel

For the record we note the obvious

Prop osition The section is parallel if and only if the connection is

fsecg

preserving



The next theorem asso ciates a unique qspray with each general connec

tion on M

Theorem For each connection H on M there is an induced qspray Q

fcisg

given by

Qv v

H

v

where TM M is the natural projection and v TM We write H Q

to denote this relationship

Parallelism and

Pro of As in the rst paragraph of Poors pro of of p it is

easily veried that Q so dened is a SODE Indeed Q is a section of by

construction and Q is a section of b ecause H is a subbundle with resp ect

T

to

T

It is clear that this Q is horizontal so compatible with the given connec

tion and that it vanishes on the section of TM thus is a qspray



Denition The integral curves of the asso ciated qspray Q the Q

fcgg

geo desics are called the geodesics of the connection H

Thus Q b eing a qspray means that constant curves ct p M for all t

are degenerate geo desics a standard prop erty of traditional geo desic sprays

of metric tensors

Denition A connection H with H Q is complete if and only if Q

fccg

is complete as a vector eld on TM

Reckziegel and Wilhelmus dene a complete connection to b e one for

which every path has a horizontal lift This conicts sharply with the usual

and our use of complete for connections

For connections on M the semidirect pro duct structure of the group A

H

b ecomes

I GL

n

gl I GL

n

n

Given a qspray Q the connections that are compatible with Q are those

whose rst column of the blo ck from M gl is all zeros This yields an

nn

n

Q

TTM G TTM whose b ers are p encils of p ossible ane subbundle G

H

H

horizontal subspaces

Remark It follows that the space ConnM of connections on M b ers

fcfqg

over the space QSprayM of qsprays on M Since QSprayM is an ane

space then it is contractible a homotopy p oint Thus ConnM is a trivial

bundle over QSprayM by Theorem

Taking E TM Theorem allows us to dene r c by pullback as

c

r c c r c c D

c D ~

Now the geo desic equation lo oks like the usual one

Theorem A curve c in M is a of H if and only if r c

c

fgeg

Chapter Connections

Pro of For H Q we compute

r c c D

c ~

Kc D H c D

~ c ~

Kc D Qc

~

using Example Now recall that K is a b er isomorphism



Therefore given the geo desic equation as

k k i

c c c

i

i k k

c c gives the unique qspray induced by H or r then Q c

i

Denition These connection co ecients provide a sp ecic connection

flccg

c c

H asso ciated to the qspray Q We call H the LC connection of Q

Here LC stands for LeviCivita as these connections will play a similar role

for qsprays

Finally we pick up the last thread from the previous section and move

toward torsion

Theorem Two connections H and H have the same geodesic qspray

fadog

if and only if the associated dierence operator is alternating

v and Pro of For each v TM Qv

H

v

v J Dv v Qv v

v

H H

v v

Therefore Q Q if and only if D vanishes on the diagonal of TM TM



Note that if b oth connections are linear then D is bilinear and skewsymmetric

c

Given a connection H with induced qspray Q let H b e the LC connection

asso ciated to Q as in Denition

Denition These LC connections are the torsionfree connections

ftfcg

c

Now let H b e a connection on M Q its induced qspray and H the

asso ciated LC connection of Q

b

Denition For D r r the torsion of H is T D

ftrsg

The factor of is included so that this will b ecome the usual torsion when b oth

b

T connections are linear and to recover similarly the usual formula r r

Automorphisms

Remark By Remark the space of connections on M b ers over

ftfg

the space of quasisprays The previous results show that torsion is what varies

among connections along each b er Thus our general torsion plays the same

role for general connections that classical torsion do es for linear connections

Automorphisms

Let Di M denote the dieomorphism group of M and Di M Ob

serve that TM TM and T M T M are vectorbundle auto

morphisms the latter for the structure only Further note that any

T

automatically restricts to a vectorbundle automorphism of V

Let H denote a general connection over M

Denition The automorphism group of H over M is

fagcg

Aut H f Di M j r rg

M

for r the unique covariant derivative corresp onding to H

We also denote this group by Aut r

M

When r is linear this coincides with the socalled ane transformation

group of r p Prop a Lie group naturally endowed with

the Schwartz top ology

Lemma If we regard H as a vector subbund le of T M then

fagcg

Aut H f Di M j H H g

M



Note that we have not required that j b e the identity map on H Of

H

course it must map b ers to b ers and b e a vectorspace isomorphism of each

b er equivalently it must b e a vectorbundle automorphism of H over TM

Lemma If we regard the eld of horizontal projections H as a section

fagcg

of EndT M over TM then

Aut H f Di M j H H g

M



One can use this to make yet another alternative denition of Aut H

M

The p oints here are that H is a tensor on TM and that the condition on

is familiar H is related to itself

Chapter Connections

Remark Now Aut H is a subgroup of Di M and should b e a

M

lo cally convex Lie group Its lo cally convex Lie algebra aut H should

M

b e the set of all complete vector elds U on M such that

$

r

U

as in the linear case This should b e pursued

$

Recall that the is a morphism of Lie brackets

Theorem If r is a connection on TM with geodesic qspray Q then

fiacg

c

U is an innitesimal automorphism of r if and only if S U S

c

U

The pro of is a lengthy but simple computation which we omit The result

is this equation which one may wish to compare with the result for a linear

connection in the pro of of Theorem

k l i l i i

$

r y y y U U U

U

k j l l

j j

fqkeg

i k l i l

y y y U U

j

l k

j

l

This leads us to call such vector elds U quasiKil ling elds they are Killing

elds when the connection is a metriccompatible linear connection for some

pseudoRiemannian metric p

j i i

y y aggre Remark What happ ens here is that the qspray Q

j

fdccg

gates the connection comp onents so that the same qspray is induced by all

general connections that dier only in torsion But taking a Lie bracket

with Q at least partially disaggregates those connection comp onents enough

so that the result dep ends on the original connection that induced Q

more results

Exponential Maps

There are two basic ideas b ehind exp onential maps in geometry One comes

from the exp onential map in Lie theory suggesting a lo cal dieomorphism

from TM to M The other comes from the idea of following a ow for a xed

globally uniform time

Combining these ideas suggests lo oking at vector elds on TM It will

turn out that following a ow in TM and then pro jecting to M works b est

for quasisprays

better explanation here

However we must give up the xed globally uniform time value if we wish to

handle inhomogeneous quasisprays

Denition and construction

We b egin with an example due to J Heb da that validates the last remark

Example Consider the SODE on R given by

fhebdag

x x

Rewriting as

dx

dt

x

we obtain

x t tan t C x tan C

so that

xt log jsec t C j C

For C x cannot b e extended b eyond

C

t

Chapter Exponential Maps

One should regard this as lo cal near x and then repro duced in charts all

along the xaxis

Note that if a SODE is homogeneous then we may extend all integral

curves to at least globally on M

Let Q b e a qspray over M We shall dene general exp onential maps

plural for Q in three steps

First let p M v T M and c the unique Qgeo desic such that

p

ct Qc t

c p

c v

v c for all v T M for which this makes sense From Dene exp

p

p

v is well dened for all in some the FEUT et al it follows that exp

p

interval dep ending on p and for v in some op en neighborho o d U of

p p p

at each p M T M dep ending on p and This denes exp

p p p

p

Next choose a smo oth function F such that p for all p M

p

will b e smo oth in it is already smo oth Cho osing smo oth means that exp

p

in all other parameters Making is the standard convention

Lastly we pro duce the global maps

Denition Dene the global exponential maps p ointwise by

fexpg

p

exp exp

p

p

for all p M

Note that there is a dierent exp onential map for each choice of such a smo oth

function M

Supp ose L M is a submanifold ie an injective immersion For

each p L dene the normal bund le b erwise via

N L T M T L

p

p p

Suppressing and regarding L M we abbreviate this as N L T M T L

p p p

S

Then NL N L is the normal bundle to L in M

p

pL

Denition A lo cal tubular neighborhood of L in M is a lo cal embed

ftblnbdg

ding NL M such that j where denotes the section of NL

and the zero section maps to L as the inclusion into M

Ex Unsuppress and in the denition of tubular neighborho o d

Definition and construction

Ex If L is embedded the tubular neighborho o d can b e made global

Observe that the domain of exp is a tubular neighborho o d of the section

in TM and that the graph of is in a tubular neighborho o d of the section

of the trivial line bundle M R

The closer the graph of gets to the section of M R the larger the

tubular neighborho o d of the section in TM b ecomes

1 2

Prop osition For we have domexp domexp attaining

fdoxg

al l of TM when and exp

Pro of

to be done



Note that we nd the bundle pro jection TM M in the interesting

p osition of b eing a member of a parameter family of smo oth maps all

other members b eing lo cal dieomorphisms

Theorem For every such that jj the general exponential

p

feldg

is a dieomorphism of an open neighborhood of T M and an map exp

p

p

open neighborhood of p M

Pro of We slightly generalize the usual pro of for sprays based on the FEUT

v etc as found for example in p f Note that for v T M exp

p

p

v where is the lo cal ow of Q Then on the section of TM

the induced tangent map exp in blo ck form lo oks like

A

I I

where A is invertible When Q is homogeneous and then A I also

as usual

expand this



Lemma In exp v the is a geodesic parameter

p



fgpg

Chapter Exponential Maps

av Each black curve is a geo desic with and Figure curves exp

p

a and v xed From shortest to longest in each plume a steps in increments of

from to In each plume v is constant There are three implicit aparameter

curves readily lo cated one along the endp oints of each of the three plumes

fjg

av In general a is not a geo desic parameter See Now consider exp

p

Figures and for a comparison Also note that these aparameter curves

are the exp onentials of radial lines in T M

p

Prop osition If Q is homogeneous then a is a geodesic parameter

 fagpg

i i

Example In R consider the qspray given by S x y y for i

fevag

t

The geo desics are readily computed as ct v e p with initial p osition

p and initial velocity v Since these are dened for t we may use the

usual exp onential map to obtain exp v c v e p in R

p

For the aparameter curves we nd exp av av e p Thus the geo desics

p

have exp onential growth in the geo desic parameter t while the acurves have

The APS correspondence

av This is one plume from Figure Each black curve Figure curves exp

p

is a geo desic and each gray blue curve is an aparameter curve The new Jacobi

elds are along the black curves but tangent to the gray curves

fjg

only linear growth in a Since these growth rates are dierent S is an inho

mogeneous qspray

Finally we recall the unique natural decomp osition of an arbitrary SODE

on TM into the sum of a quasispray and the vertical lift of a vector eld on M

v

Sec Set R u J KS for u T M and p M Then R W

u p

for some W X Q S R is a quasispray and

S Q R fqrsg

is the natural decomposition of S

The APS corresp ondence

We rst recall the classical situation

Denition A quasispray S is called a spray if and only if S av

fsprayg

a aS v for all v TM

In induced lo cal co ordinates S x y x y ay a S y whence it is a

vector eld that is hd on TM Such a vector eld is said to b e quadratic

Chapter Exponential Maps

classical APS

Theorem general AmbrosePalaisSinger Given a quasispray Q

fapsg

on M there exists a compatible connection H in TTM

The idea of the pro of is standard use Q and its exp onential maps to dene

notions of horizontal and parallel that will coincide with the usual ones along

a path for any H Q The main problem is that rays do not necessarily

exp onentiate to Qgeo desics

Pro of We give an explicit construction Let denote the lo cal ow of Q in

TM and an integral curve of Q with v T M

p

v

For each v T M choose an so that exp v is dened For t

p v v

p

dene

t

v

v exp t exp

v

p p

Then v and exp onentiates to the Qgeo desic with

v v v v

initial condition v at p Note that if Q is homogeneous then t tv

v

Now for each w T M dene

p

d

H v T M J w

w p

t

v

dt

t

Clearly this do es not dep end on any choice of If Q is a spray then exp tv

v

p

tv

It remains to verify that this H is indeed a connection and that H Q

to be done Poor pp proof of



Corollary This constructed H Q is the LC connection of Q

flccg 

Jacobi elds

Geo desic variational vector elds are dened much as in and the geo desic

variational or Jacobi equations written as in

Denition Let c I M b e an S geo desic for a SODE S over M A

fgvg

geodesic variation of c is a smo oth map I M such that

t ct for all t in I and

Jacobi fields

s is an S geo desic for each s in

Then the variational vector eld V j x y in induced lo cal

s s

co ordinates on TM is a solution of the following system

i i

S S

i j j

y fgveg x y x y

j j

y x

We refer to this system as the variational equations of the geo desic equation

Note that this requires only a SODE S no connection is needed yet

We change the denition of Jacobi eld from the usual to remove the

historical artifacts of metric tensors and connections

Denition A Jacobi eld along a geo desic is a geo desic variational

fjfg

vector eld of and along it

Conjugate p oints are dened much as usual

Denition Let b M b e a geo desic segment with p

fcpg

and u Then b is conjugate to p along if and only if there exists

a nontrivial Jacobi eld V along such that V V b

By nontrivial we mean of course not identically zero

We mo dify the statement of Theorem of for use here

Theorem Let S be a SODE over M fjfgg

c

S is a SODE over TM

c

V is a geodesic of S in TM if and only if V is a Jacobi eld along

V in M

The domains of the geodesic ows are related by

c

dom S dom S

R

Pro of

to be done Bucataru Dahl 

Chapter Exponential Maps

We shall show that conjugate p oints are precisely the p oints where exp o

nential maps drop rank following as a general outline The rst result is

an immediate consequence of the preceding theorem and recovers Lem

p

Corollary Given u v T M and an S geodesic segment b M

p

fujfg

with p there exists a unique Jacobi eld V along such that V u

and V J v

u



In the next result we use to denote the unique inextendible geo desic

u

with initial velocity u T M as in and to denote the unique curve

p u

from to u T M that exp onentiates to a geo desic segment in M as in

p

the pro of of Theorem This gives us the relevant parts of Prop

p Note that we must now assume that S is a quasispray Q The natural

decomp osition shows that this is not a real restriction

Prop osition If u v T M and V is the unique Jacobi eld along

p u

fexpjfg

such that V and V J v then

u

J v V exp

u

p

for each suciently smal l

t

u sv t is a geo desic variation of when Pro of Note that exp

usv u

p

t and s is suciently small Then the variational vector eld is

j t V t exp

s s usv

p

and is a Jacobi eld by denition

Recall from the pro of of Theorem that and

usv usv

J v Using the pro of of Theorem u sv Then V and V exp

u

p

we further obtain V J v

u



Now we can obtain the desired result as in Prop p

Theorem Conjugate points are where the exponential maps drop rank

fcpdrg

Pro of Let b M b e a geo desic segment with p and u

Then b is conjugate to p if and only if there exists a nontrivial geo desic

variational vector eld V along such that V V b moreover

V J v for some v T M Note that b oth u and v must b e nonzero

u p

b

Then from the preceding prop osition it follows that exp J v

u

p

Jacobi fields

t

Conversely with u and v as b efore exp u sv provides the desired

p

geo desic variation of and from the pro of of the preceding prop osition we

obtain a Jacobi eld along as required



If a connection H Q then it is now reasonably straightforward to get the

traditional Jacobi Equation for Jacobi elds in the form

fjeg V R V

Note that this equation is linear in V although not in Thus there is as

usual a vector space of Jacobi elds along of dimension n by Corollary

We may also consider the Jacobi operator R R and represent

it by a nsquare matrix as usual If all its eigenvalues are negative we have

Jacobi stability of the geo desic system near

For complete clarity we maintain a careful distinction b etween vertical

vectors tangent to a b er in TM and tangent vectors in a b er of TM It

seems to b e most common however esp ecially in older texts to b e quite lib eral

in identifying vertical and tangent vectors and suppressing the identications

as much as p ossible

An example is the prime notation as used by ONeill among others

0

V r V

So the Jacobi equation b ecomes

V R V

Recall that unlike ONeill but like Poor we dene and thereby justify r V

by a pullback rather than by an extension so ONeills caveat p do es not

apply here

more results

Our results in this section provide the foundation for further exploration

of situations involving Jacobi elds or the Jacobi op erator such as Jacobi

stability and the Osserman condition

Gconnections

Some introduction

Induced connections

Theorem Every Gconnection on an associated Gbund le is induced by

fgcpcg

a principal Gconnection

Pro of

to be done



Linear Connections

Some introduction

Automorphisms

$

Recall that the Lie derivative is a morphism of Lie brackets

Theorem If r is linear with geodesic spray S then U is an innites

fialcg

c

imal automorphism of r if and only if S U S

c

U

Pro of We use induced lo cal co ordinates x y X Y and calculate the Lie

c

j j k i i i

bracket of U x y U y are y y where the U and S x y y

j

j k j k

the connection comp onents and denotes the lifted partial with resp ect to

i

i i

x the partial with resp ect to x

terms First we compute the

i

i j i j j i

U y y U y U

i i j j i

j j k l i l i j i

y y y U U y y U

i j i j i

l

j k

The X comp onent is the sum of these which vanishes identically Thus the

c

Lie bracket U S is a vertical vector eld on TM

Now for the terms First the XY crisscross terms

i

i l j k i j k

U y y y y U

i i j

l k

j k

i j k k j i l

y y y y U U

j i

l k

j k

l i j k l i j k l i j k

U y y U y y U y y

l l l

j k j k j k

k j i k j i

y y U y y U

j j i

k k

Observe that in the last expression the second third and fourth terms vanish

automatically This leaves us with

i j k k j i l i j k k j i l

y y y U U y y y U y y U

j i i

l k l k j

j k j k

Chapter Linear Connections

l i i j k

U U y y fag

i

l k j

j k

Next the Y Y terms

j l i j k i j k l i

y U y y y y y U

j i i i

l l

j k j k

l j i k l k i j l j k i

y U y y U y y y U

i

l l l

j k j k j k

l i l i l i j k

U U U fag y y

j i

k l

l k j l j k

$

Combining and we obtain the condition for S or

c

U

for U to b e an innitesimal automorphism of r

l i l i i i l i l

U U U U U

j

k j l l k

j k j k l k j l

Taking into account that these connection co ecients are the negatives of

those in this is merely equation on p there expressing the con

$

dition that r

U



Apparently this result rst app eared as an exercise in p alb eit

with a much more complicated pro of We chose our pro of precisely b ecause it

is simple and nave alb eit tedious

j k i i

y y Remark What happ ens in the theorem is that the spray S

j k

fdlccg

aggregates the connection comp onents so that the same spray is induced by

all linear connections that dier only in torsion But taking a Lie bracket with

S at least partially disaggregates those connection comp onents enough that

the result dep ends on the original connection that induced S

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Index

Natural intraentry ordering is preferred reverse ordering is deprecated

preserving curvature

form

absolutely parallel

op erator

ane

dierence op erator

bundle

connection

Ehresmann

angle b etween

connection

automorphism

Gconnection

group of a connection

exp onential maps

biinvariant

at connection

structure

fundamental vector eld

canonical

Gconnection

connection

general connection

structure

geo desic

Cartan connection

equation

chain rule

variation

conjugate p oint

globally

connected

parallel

group oid

Grassmann bundle

connection

Grassmannian

co ecients

group oid

connector

contractible

holonomy

covariant

group

derivative

group at a p oint

dierential morphism

Index

parallel holonomy bundle

along homotopic

transp ort homotopy

principal equivalent

connection inverse

pseudoRiemannian connection p oint

pullback type

bundle horizontal

connection bundle

connection along Grassmann bundle

Grassmannian

Qgeo desics

lift

quadratic vector eld

inessential

restricted holonomy group

innerpro duct bundle

Riemannian

bundle Jacobi

inner pro duct equation

rightinvariant eld

structure op erator

stability

spray

standard Koszul connection

connection

LC connection

structure

leftinvariant

strongly nonlinear connection

structure

symmetric

Lies First Theorem

connection

lift

connector

linear

tensorial connection

torsion lo op group

torsionfree connection Lorentzian inner pro duct

trivial

nonlinear connection

connection

normal bundle

tubular neighborho o d

nullhomotopic

uniformly vertically b ounded

oriented

UVB

path space

variational Osserman condition

Index

equations

vector eld

vertex group

vertically constant

Wong angles