# Lie groups and Chern-Simons Theory

Benjamin Himpel  Reutlingen University
Department of Informatics
Alteburgstr. 150
72762 Reutlingen

###### Abstract.

Witten introduced classical Chern-Simons theory to topology in 1989, when he
defined an invariant for knots in 3-manifolds by an integral over a certain
infinite-dimensional space, which up to today have not been entirely
understood. However, they motivated lots of interesting questions and results
in knot theory and low-dimensional topology, as well as the development of
entirely new fields.

These are lecture notes for a course in Lie groups and Chern-Simons Theory
aimed at graduate students.

###### Contents

  1. 1 Introduction
    1. 1.1 An overview
  2. 2 Background material
    1. 2.1 Lie groups
    2. 2.2 Lie algebras
    3. 2.3 Homomorphisms
    4. 2.4 Lie subgroups
    5. 2.5 The exponential map
    6. 2.6 Lie Group actions
    7. 2.7 Principal bundles
    8. 2.8 Homology with local coefficients
    9. 2.9 Algebraic sets and the Zariski tangent space
  3. 3 Connections on principal bundles
    1. 3.1 A few equivalent notions
    2. 3.2 Structure of the space of connections
    3. 3.3 Lifting paths
    4. 3.4 Connections in vector bundles
    5. 3.5 Equivalence of connections in vector bundles and principal bundles
    6. 3.6 Extending the covariant derivatives
    7. 3.7 The Maurer-Cartan form
    8. 3.8 Curvature
    9. 3.9 Gauge transformations
    10. 3.10 Flat connections and the fundamental group
    11. 3.11 Chern-Weil Theory
    12. 3.12 The Chern-Simons form
  4. 4 Chern-Simons theory
    1. 4.1 Principal bundles on 3–manifolds
    2. 4.2 The Chern-Simons function
    3. 4.3 An Euler characteristic of the gauge equivalence classes of $\text{\it{SU}}(2)$–connections
    4. 4.4 Classical Chern-Simons theory

##  Chapter 1 Introduction

These are notes for a course in an active field of geometric topology at the
border to mathematical physics. They are aimed at graduate students, who know
the basics of differentiable manifolds and differential forms. It is based on
parts of:

  * •

Frank W. Warner: Foundations of Differentiable Manifolds and Lie Groups [62].

  * •

Dan Freed: Classical Chern-Simons Theory, Part 1 [21].

  * •

Simons Donaldson and Peter Kronheimer: The Geometry of Four-Manifolds [19].

  * •

Michael Bohn: On Rho invariants of fiber bundles [12]

  * •

Unpublished notes by Paul Kirk.

After an introduction to Lie groups and Lie algebras, principal bundles,
connections and gauge transformations, we will carefully construct the Chern-
Simons action and study the moduli space of its classical solutions. This
yields Taubes’ beautiful and influential description of Casson’s invariant for
homology 3-spheres via Chern-Simons theory [59], for which we need such
concepts as differential operators and spectral flow. This naturally leads to
subjects like the eta invariant and the rho invariant on the one hand as well
as the quantization of the Chern-Simons action and Witten’s invariants on the
other.

###  1.1. An overview

Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ its Lie algebra. A
connection $A$ on the trivial $G$-bundle over a closed oriented 3–manifold $M$
is a $\mathfrak{g}$-valued 1–form, i.e. $A\in\Omega^{1}(M;\mathfrak{g})$.
Chern-Simons theory is a quantum field theory in three dimensions, whose
action is proportional to the Chern-Simons invariant given in [17]

$cs(A)=\frac{1}{8\pi^{2}}\int_{M}\operatorname{{tr}}(A\wedge
dA+\frac{1}{3}A\wedge[A\wedge A]).$

Witten introduced Chern-Simons theory to knot theory in 1989 [64], when he
described for each integer level $k\in{\bf Z}$ an invariant of a link
$L=(L_{j})$ in a 3–manifold $M$ (and a list of finite-dimensional
representations $\rho_{j}$ of $G$ associated to $L_{j}$) as the (non-rigorous)
Feynman path integral

$Z_{k}(M,L)=\int_{{\mathcal{A}}/{\mathcal{G}}}e^{2\pi
ki\;cs(A)}\prod_{j}\operatorname{{tr}}_{\rho_{j}}(\operatorname{hol}_{A}(L_{j}))dA,$

where ${\mathcal{A}}$ is the space of $G$–connections, ${\mathcal{G}}$ is the
space of gauge transformations. He interpreted these invariants using the
axioms of topological quantum field theory (TQFT) as well as via an asymptotic
expansion —the semiclassical approximation— by using the method of stationary
phase. There have been several advances in understanding both approaches
separately, notably the rigorous construction of the TQFT version by
Reshetikhin and Turaev [51], the first computer calculations and the
refinement of the semiclassical approximation by Freed and Gompf [22] and
various recent work by Andersen, Hansen, Ueno and Takata [3, 5, 6, 4, 29, 28,
30, 31] in the realm of the asymptotic expansion conjecture for the
Resehetikhin-Turaev invariants [48, Section 7.2]. However, there have been
relatively few developments relating the TQFT version to the semiclassical
approximation apart from the direct comparison for lens spaces for
$G=\text{\it{SU}}(2)$ and some torus bundles over $S^{1}$ by Jeffrey [38],
where she explicitly analyzed the asymptotic behaviour of the Reshetikhin-
Turaev invariants and exhibited agreement in the leading term with the
semiclassical approximation. One promising approach is to give a rigorous
treatment of the Feynman path integral version using stochastical analysis via
Fresnel integrals or Hida distributions, if possible, this would provide a
strong link between the TQFT version and the semiclassical approximation. See
[2, Section 10.5.5] by Albeverio, Høegh-Krohn, and Mazzucchi as well as [27]
by Hahn for an overview.

In [64], Witten proposed a physical interpretation of the Jones polynomial
using Chern-Simons theory, and [39] Kashaev made a remarkable prediction that
the volume of a hyperbolic knot $K$ in $S^{3}$ is given by the limit of the
colored Jones polynomial $J_{N}(K,q)$ of $K$. This is known as the volume
conjecture, and the precise statement is that $\log|J_{N}(K,e^{2\pi i/N})|/N$
limits to $\frac{1}{2\pi}{\rm Vol}(S^{3}\setminus K)$ as $N\to\infty.$ (This
conjecture can be extended to all knots by simply replacing hyperbolic volume
with simplicial volume on the right hand side.)

Using $\text{\it{SL}}(2,{\bf C})$ Chern-Simons theory, Gukov conjectured in
[25] that the asymptotic expansion of the colored Jones polynomial
$J_{N}(K,q)$ as $N\to\infty$ and $q\to 1$ should be equal to the partition
function of the $\text{\it{SL}}(2,{\bf C})$ Chern-Simons knot theory on
$S^{3}\setminus K$, and using this he derived a parameterized version of the
volume conjecture which postulates a relationship between the family of limits
of the colored Jones polynomial and the volume function ${\rm Vol(K,u)}$ on
the character variety of the knot complement. The paper [26] provides further
evidence for the generalized conjecture by by studying some of the sub-leading
terms in the asymptotic expansion of the colored Jones polynomial.

There is a strong connection between Witten’s invariants and the so-called
finite-type invariants. The first definition of a finite-type 3-manifold
invariant was given by Ohtsuki [49], though by now many other useful and
equivalent definitions have been given (see e.g. [23, 45]). The first
nontrivial finite type 3–manifold invariant can be identified with the
$\text{\it{SU}}(2)$ Casson invariant [1, 61] , and indeed all the terms in the
stationary phase expansion of the Chern-Simons path integral give rise to
finite type 3–manifold invariants. (See the papers [52, 44] for a
mathematically rigorous definition of invariants arising from a stationary
phase approximation, as well as proofs the invariants are of finite-type.)

A different direction of research was initiated by Taubes in 1990 [59]. He
laid the groundwork for new topological invariants motivated by Chern-Simons
theory by showing that the $\text{\it{SU}}(2)$ Casson invariant has a gauge
theoretical interpretation as the Euler characteristic of
${\mathcal{A}}/{\mathcal{G}}$ in the spirit of the Poincaré-Hopf theorem,
where he views the Chern-Simons invariant as a $S^{1}$-valued Morse function
on ${\mathcal{A}}/{\mathcal{G}}$. Taubes realized that the Hessian of the
Chern-Simons invariant and the odd signature operator coupled to the same path
of $\text{\it{SU}}(2)$ connections have the same spectral flow. Floer extended
this idea around the same time to instanton Floer homology [20], which has the
$\text{\it{SU}}(2)$ Casson invariant as its Euler characteristic, by viewing
the critical points of the Chern-Simons function as a ${\bf Z}/8$ graded Morse
complex.

The Seiberg-Witten revolution in gauge theory shifted attention from instanton
Floer homology to the more difficult monopole Floer homology, and progress in
either of these versions of Floer homology has been vastly outpaced by the
remarkable achievements in the closely related but much more accessible
Heegaard-Floer homology introduced by Ozsváth and Szabó. Heegaard-Floer
includes an entire package of homology theories which have become powerful
tools in low-dimensional topology, and there has been some success in defining
the corresponding pieces in instanton Floer and monopole Floer. Interest in
these Floer homologies has been rekindled by the work [42, 41, 43] of
Kronheimer and Mrowka; in [42] they give a thorough and rigorous treatment of
monopole Floer theory, in [41] they define knot Floer homology using
instantons, and in [43] they use sutured Floer homology to give a new and more
direct proof of property $P$ for knots, as well as the result, originally due
to Ghiggini and Ni in the Heegaard-Floer setting, that shows that monopole
Floer knot homology detects fibered knots.

In summary, Chern-Simons gauge theory is an active area of research, which has
produced a lot of interesting questions and results, as well as initiated the
development of entirely new fields.

##  Chapter 2 Background material

###  2.1. Lie groups

Lie groups are groups which are also differentiable manifolds, in which the
group operations are smooth. Well-known examples are the general linear group,
the unitary group, the orthogonal group and the special linear group.

######  Definition 2.1.1.

A Lie group $G$ is a differentiable manifold with a group structure, such that
the map $G\times G\to G$ given by $(g,h)\mapsto gh^{-1}$ is $C^{\infty}$.

######  Exercise 2.1.2.

Let $G$ be a Lie group.

  1. (1)

Show that the maps $g\mapsto g^{-1}$ and $(g,h)\mapsto gh$ for $g,h\in G$ are
$C^{\infty}$.

  2. (2)

Show that the identity component of $G$ is a Lie group.

######  Exercise 2.1.3.

Convince yourself, that the following spaces are Lie groups.

  1. (1)

${\bf R}^{n}$ under vector addition.

  2. (2)

${\bf C}\setminus\\{0\\}$ under multiplication.

  3. (3)

$S^{1}=U(1)$ under multiplication.

  4. (4)

$G\times H$ for Lie groups $G$ and $H$ with the product manifold structure and
the group structure
$(g_{1},h_{1})(g_{2},h_{2})\coloneqq(g_{1}g_{2},h_{1}h_{2})$.

  5. (5)

$T^{n}=\\{z\in{\bf C}^{n}\mid|z_{1}|=\ldots=|z_{n}|=1\\}$.

  6. (6)

The non-singular real matrices $Gl(n,{\bf R})$ under matrix multiplication.

###  2.2. Lie algebras

######  Definition 2.2.1.

A Lie algebra $\mathfrak{g}$ over ${\bf R}$ is a real vector space
$\mathfrak{g}$ together with a bilinear operator
$[\cdot,\cdot]\colon\thinspace\mathfrak{g}\times\mathfrak{g}\to\mathfrak{g}$
(the bracket) such that for all $x,y,z\in\mathfrak{g}$,

  1. (1)

$[x,y]=-[y,x]$. (anti-commutativity)

  2. (2)

$[[x,y],z]+[[y,z],x]+[[z,x],y]=0$. (Jacobi identity)

We will see that there is a Lie algebra associated to each Lie group, and that
every connected, simply connected (i.e. the fundamental group $\pi_{1}(G)$ is
trivial) Lie group is determined up to isomorphism by its Lie algebra. The
study of such Lie groups then reduces to a study of their Lie algebras.

######  Exercise 2.2.2.

Convince yourself that the following vector spaces are Lie algebras.

  1. (1)

The vector space of all smooth vector fields under the Lie bracket on vector
fields.

  2. (2)

Any algebra (e.g. all $n\times n$ matrices $\mathfrak{gl}(n,{\bf R})$) with

$[A,B]\coloneqq AB-BA.$

  3. (3)

${\bf R}^{3}$ with the cross product as the bracket.

######  Definition 2.2.3.

Let $\varphi\colon\thinspace M\to N$ be $C^{\infty}$. Smooth vector fields $X$
on $M$ and $Y$ on $N$ are $\varphi$-related if $d\varphi\circ
X=Y\circ\varphi$, which is short for $d\varphi_{p}(X_{p})=Y_{\varphi(p)}$ for
all $p\in M$.

######  Exercise 2.2.4.

Let $\varphi\colon\thinspace M\to N$ be $C^{\infty}$. Let $X_{1}$ and $X_{2}$
be smooth vector fields on $M$, and let $Y_{1}$ and $Y_{2}$ be smooth vector
fields on $N$. If $X_{i}$ is $\varphi$-related to $Y_{i}$, $i=1,2$, then
$[X_{1},X_{2}]$ is $\varphi$-related to $[Y_{1},Y_{2}]$.

######  Definition 2.2.5.

Let $G$ be a Lie group and $g\in G$. Left translation $l_{g}$ and right
translation $r_{g}$ by $g$ are diffeomorphisms of $G$ given by

$l_{g}h=gh\quad r_{g}=hg,\quad\text{for all }h\in G.$

A vector field $X$ on $G$ is called left-invariant if $X$ is $l_{g}$-related
to itself for all $g\in G$, that is

$\displaystyle dl_{g}\circ X$ $\displaystyle=X\circ l_{g},$ which is short for
$\displaystyle\quad dl_{g}(X_{h})$ $\displaystyle=X_{gh}\quad\text{for all
}h\in G.$

######  Proposition 2.2.6.

Let $G$ be a Lie group and $\mathfrak{g}$ its set of left-invariant vector
fields.

  1. (1)

$\mathfrak{g}$ is a real vector space, and the map
$\alpha\colon\thinspace\mathfrak{g}\to T_{e}G$ defined by $\alpha(X)=X(e)$ is
an isomorphism of $\mathfrak{g}$ with the tangent space $T_{e}G$ of $G$ at the
identity. Consequently $\dim(\mathfrak{g})=\dim(G)$.

  2. (2)

Left invariant vector fields are smooth.

  3. (3)

The Lie bracket of two left-invariant vector fields is a left-invariant vector
field.

  4. (4)

$\mathfrak{g}$ is a Lie algebra under the Lie bracket operation of vector
fields.

###### Proof.

It is not difficult to see that $\mathfrak{g}$ is a real vector space and that
$\alpha$ is linear. Since $\alpha(X)=\alpha(Y)$ yields

$X_{g}=dl_{g}(X_{e})=dl_{g}(\alpha(X))=dl_{g}(\alpha(Y))=dl_{g}(Y_{e})=Y_{g}\quad\text{for
each $g\in G$,}$

$\alpha$ is injective. In order to see the surjectivity of $\alpha$ let $v\in
T_{e}G$ and define

$X_{g}=dl_{g}(v)\quad\text{for all }g\in G.$

Then $\alpha(X)=v$ and the left-invariance of $X$ is shown by

$X_{gh}=dl_{gh}(v)=dl_{g}dl_{h}(v)=dl_{g}(X_{h})\quad\text{for all }g,h\in G.$

This proves part (1).

For (2) let $X\in\mathfrak{g}$ and $f\in C^{\infty}(G)$. We need to show that
$Xf\in C^{\infty}(G)$. Let $\varphi\colon\thinspace G\times G\to G$ be the
smooth map given by multiplication $\varphi(g,h)=gh$. Let $i^{1}_{e}$ and
$i^{2}_{g}$ be the smooth maps $G\to G\times G$ given by $i^{1}_{e}(h)=(h,e)$
and $i^{2}_{g}(h)=(g,h)$. Let $Y$ be any smooth vector field on $G$ with
$Y_{e}=X_{e}$. Then $[(0,Y)(f\circ\varphi)]\circ i_{e}^{1}$ is smooth and

$\displaystyle[(0,Y)(f\circ\varphi)]\circ i_{e}^{1}(g)$
$\displaystyle=(0,Y)_{(g,e)}(f\circ\varphi)$
$\displaystyle=0_{g}(f\circ\varphi\circ i^{1}_{e})+Y_{e}(f\circ\varphi\circ
i^{2}_{g})$ $\displaystyle=X_{e}(f\circ\varphi\circ i^{2}_{g})=X_{e}(f\circ
l_{g})$ $\displaystyle=dl_{g}(X_{e})f=X_{g}f=Xf(g),$

which proves part (2).

Since by (2), left-invariant vector fields are smooth, their Lie brackets are
defined and by Exercise 2.2.4 they are again left-invariant, which shows (3).
(4) follows from (3) and part (1) of Exercise 2.2.2 ∎

######  Definition 2.2.7.

We define the Lie algebra of the Lie group $G$ to be the Lie algebra
$\mathfrak{g}$ of left-invariant vector fields on $G$.

Often it will be convenient to think of the $T_{e}G$ as the Lie algebra of $G$
with the Lie algebra structure induced by the isomorphism $\alpha$ from
Proposition 2.2.6(1).

######  Example 2.2.8.

The real line ${\bf R}$ is a Lie group under addition. The left-invariant
vector fields are simply the constant vector fields
$\\{\lambda(\frac{d}{dr})\mid\lambda\in{\bf R}\\}$. The bracket of any two
such vector fields is 0.

######  Example 2.2.9.

The Lie group $\text{\it{GL}}(n,{\bf R})=\det^{-1}({\bf R}\setminus\\{0\\})$
is a (differentiable) submanifold of $\mathfrak{gl}(n,{\bf R})$.
$\text{\it{GL}}(n,{\bf R})$ has global coordinates $A_{ij}$ which assigns to
each matrix $A$ the $ij$-th entry. Since $\det(A^{-1})=\det(A)^{-1}$ and
$\det(AB)=\det(A)\det(B)$ for any $A,B\in\text{\it{GL}}(n,{\bf R})$, the
matrix $AB^{-1}$ is invertible with the $ij$-th entry $(AB^{-1})_{ij}$ being a
rational function in the entries of $A$ and $B$ with non-zero denominator.
Therefore $(A,B)\mapsto AB^{-1}$ is $C^{\infty}$.

Consider the isomorphism $\alpha\colon\thinspace\mathfrak{g}\to T_{e}G$ from
Proposition 2.2.6(1). Since $T_{e}\text{\it{GL}}(n,{\bf
R})=T_{e}\mathfrak{gl}(n,{\bf R})$ and the map $\beta\colon\thinspace
T_{e}\mathfrak{gl}(n,{\bf R})\to\mathfrak{gl}(n,{\bf R})$ given by
$\beta(v)_{ij}\coloneqq v_{ij}$ is a (canonical) isomorphism,
$\beta\circ\alpha$ induces a Lie algebra structure on $\mathfrak{gl}(n,{\bf
R})$. We leave it as an exercise to show the bracket agrees with the usual
$[A,B]=AB-BA$.

######  Exercise 2.2.10.

Let $\mathfrak{g}$ be the Lie algebra of $\text{\it{GL}}(n,{\bf R})$. Prove
that $\beta\circ\alpha$ from Example 2.2.9 induces the correct Lie algebra
structure, i.e. show that for $X,Y\in\mathfrak{g}$ we have
$\beta\circ\alpha([X,Y])=[\beta\circ\alpha(X),\beta\circ\alpha(Y)]$.

Taking Example 2.2.9 as a starting point, we can create other examples: The
non-singular matrices $\text{\it{GL}}(n,{\bf C})$ of all $n\times n$ complex
matrices $\mathfrak{gl}(n,{\bf C})$ is a Lie group with Lie Algebra
$\mathfrak{gl}(n,{\bf C})$. If $V$ is an $n$-dimensional real vector space, a
basis of $V$ determines a diffeomorphism from $\operatorname{End}(V)$ to
$\mathfrak{gl}(n,{\bf R})$ sending $\operatorname{Aut}(V)$ onto
$\text{\it{GL}}(n,{\bf R})$. In this way $\operatorname{End}(V)$ is a Lie
group with Lie algebra $\operatorname{Aut}(V)$. We get an analog example for
complex case vector spaces.

The special linear group $\text{\it{SL}}(n,{\bf C})$, the unitary group
$U(n)$, the special unitary group $\text{\it{SU}}(n)$ are the most important
examples for us.

######  Exercise 2.2.11.

Assume that the bracket corresponds to the usual bracket on matrices as in
example 2.2.9, and show the following by differentiating the relations
defining the Lie group:

  1. (1)

The Lie algebra $\mathfrak{sl}(n,{\bf C})$ of $\text{\it{SL}}(n,{\bf C})$
consists of all $n\times n$–matrices with vanishing trace.

  2. (2)

The Lie algebra $\mathfrak{u}(n)$ of $U(n)$ consists of all skew-symmetric
$n\times n$–matrices.

  3. (3)

The Lie algebra $\mathfrak{su}(n)$ of $\text{\it{SU}}(n)$ consists of all
skew-symmetric, traceless $n\times n$–matrices.

###  2.3. Homomorphisms

######  Definition 2.3.1.

A map $\varphi\colon\thinspace G\to H$ is a (Lie group) homomorphism if
$\varphi$ is both $C^{\infty}$ and a homomorphism of groups. We call $\varphi$
an isomorphism if $\varphi$ is also a diffeomorphism.
If $\mathfrak{g}$ and $\mathfrak{h}$ are Lie algebras, a map
$\psi\colon\thinspace\mathfrak{g}\to\mathfrak{h}$ is a (Lie algebra)
homomorphism if it is linear and preserves brackets
($\psi[X,Y]=[\psi(X),\psi(Y)]$ for all $X,Y\in\mathfrak{g}$). If $\psi$ is
also a bijection, then $\psi$ is an isomorphism.

Let $\varphi\colon\thinspace G\to H$ be a homomorphism. Then $\varphi$ maps
the identity of $G$ to the identity of $H$, and the differential $d\varphi$ of
$\varphi$ is a linear transformation of $\mathfrak{g}=T_{e}G$ into
$\mathfrak{h}=T_{e}H$. Notice that by the natural identification between Lie
algebras and left-invariant vector fields $d\varphi(X)$ is the unique left-
invariant vector field satisfying

(2.3.1) $(d\varphi(X))_{e}=d\varphi_{e}(X_{e}).$

######  Theorem 2.3.2.

Let $G$ and $H$ Lie groups with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$
respectively, and let $\varphi\colon\thinspace G\to H$ be a homomorphism. Then

  1. (1)

$X$ and $d\varphi(X)$ are $\varphi$-related for each $X\in\mathfrak{g}$.

  2. (2)

$d\varphi\colon\thinspace\mathfrak{g}\to\mathfrak{h}$ is a Lie algebra
homomorphism.

###### Proof.

The left-invariant vector fields $d\varphi(X)$ and $X$ are $\varphi$-related,
because

$(d\varphi(X))_{\varphi(g)}=dl_{\varphi(g)}d\varphi_{e}(X_{e})=d(l_{\varphi(g)}\circ\varphi)X_{e}=d(\varphi\circ
l_{g})X_{e}=d\varphi(X_{g}).$

To show part (2), let $X,Y\in\mathfrak{g}$. By Exercise 2.2.4 $[X,Y]$ is
$\varphi$–related to the left-invariant vector field
$[d\varphi(X),d\varphi(Y)]$, that is

$(d\varphi([X,Y]))_{e}\stackrel{{\scriptstyle\eqref{identification}}}{{=}}d\varphi_{e}([X,Y]_{e})=[d\varphi(X),d\varphi(Y)]_{\varphi(e)}=[d\varphi(X),d\varphi(Y)]_{e}$

Therefore using the identification of left-invariant vector fields with
tangent vectors at the identity,
$d\varphi([X,Y])=[d\varphi(X),d\varphi(Y)]$.111We had to be a bit careful not
to mix up notation or use short cuts: By the way we had defined
$\varphi$-related in Definition 2.2.3, we had to take the detour via the
tangent space at $e$. ∎

###  2.4. Lie subgroups

######  Definition 2.4.1.

$(H,\varphi)$ is a Lie subgroup of the Lie group $G$ if

  1. (1)

$H$ is a Lie group;

  2. (2)

$(H,\varphi)$ is a submanifold of $G$;

  3. (3)

$\varphi\colon\thinspace H\to G$ is a Lie group homomorphism.

$(H,\varphi)$ is called a closed subgroup of $G$ if $\varphi(H)$ is also a
closed subset of $G$.

Let $\mathfrak{g}$ be a Lie algebra. A subspace
$\mathfrak{h}\subset\mathfrak{g}$ is a subalgebra if $[X,Y]\in\mathfrak{h}$
for all $X,Y\in\mathfrak{h}$.222A Lie algebra $\mathfrak{g}$ or a subalgebra
of $\mathfrak{g}$ is not necessarily an algebra.

Let $(H,\varphi)$ be a Lie subgroup of $G$, and let $\mathfrak{h}$ and
$\mathfrak{g}$ be their respective Lie algebras. Then by Theorem 2.3.2
$d\varphi$ yields an isomorphism between $\mathfrak{h}$ and the subalgebra
$d\varphi(\mathfrak{h})$ of $\mathfrak{g}$.

We will show in this section one of the fundamental theorems in Lie group
theory, which asserts that there is a 1:1 correspondence between connected Lie
subgroups of a Lie group and subalgebras of its Lie algebra. We first need the
two following propositions.

######  Proposition 2.4.2.

Let $G$ be a connected Lie group, and let $U$ be a neighborhood of $e$. Then

$G=\cup^{\infty}_{n=1}U^{n}$

where $U^{n}$ consists of all $n$–fold products of elemets of $U$.

###### Proof.

Consider $V\coloneqq U\cup U^{-1}$ (where $U^{-1}=\\{g^{-1}\in U\mid g\in
U\\}$) and let

$H\coloneqq\bigcup_{n=1}^{\infty}V^{n}\subset\bigcup_{n=1}^{\infty}U^{n}.$

$H$ is a subgroup of $G$ and open in $G$, since $h\in H$ implies $hV\in H$.
Therefore each coset of $H$ is also open in $G$. Since

$H=G\setminus\bigcup_{g\in G,gH\neq H}gH,$

$H$ is also closed. Since $H$ is non-empty and $G$ is connected we get $G=H$.
∎

Recall that a $k$–dimensional distribution ${\mathcal{D}}$ on a manifold $M$
is a choice of a $k$-dimensional linear subspace ${\mathcal{D}}_{p}\subset
T_{p}M$. A distribution is smooth, if it is locally spanned by smooth vector
fields $X_{1},\ldots,X_{k}$, or equivalently if ${\mathcal{D}}$ is a smooth
subbundle of $TM$. A smooth distribution is involutive if
$[X,Y]\in{\mathcal{D}}$ for any smooth vector fields $X,Y\in{\mathcal{D}}$. It
is completely integrable, if for each point $p\in M$ there is an integral
manifold $N$ of ${\mathcal{D}}$ passing through $p$, where $N$ is integral if

$N\stackrel{{\scriptstyle i}}{{\hookrightarrow}}M\quad\text{and}\quad
di_{p}(T_{p}N)={\mathcal{D}}_{p}\quad\text{for each }p\in N.$

A smooth coordinate chart $(U,\varphi)$ is flat for ${\mathcal{D}}$, if
$\varphi(U)=U^{\prime}\times U^{\prime\prime}\subset{\bf R}^{k}\times{\bf
R}^{n-k}$, and at points of $U$, ${\mathcal{D}}$ is spanned by the first $k$
coordinate vector fields $\frac{\partial}{\partial
x^{1}},\ldots,\frac{\partial}{\partial x^{k}}$. This implies that each slice
of the form $x^{k+1}=c^{k+1},\ldots,x^{n}=c^{n}$ for constants $c^{i}$ is an
integral manifold of ${\mathcal{D}}$. Beware that the terminology is
inconsistent among various books. However, we will show the famous Frobenius’
Theorem which states that terminology is of no consequence. Before that we
need to recall a few definitions and prove a few lemmas.

######  Definition 2.4.3.

Let $M$ be a smooth manifold and let $X$ be a vector field on $M$.

  1. (1)

An integral curve of $X$ is a smooth curve $\gamma\colon\thinspace J\to M$
with $J\subset{\bf R}$ an open interval such that

$\gamma^{\prime}(t)=X_{\gamma(t)}\quad\text{ for all }t\in J.$

If $0\in J$, then $\gamma(0)$ is the starting point.

  2. (2)

Let $\theta^{(p)}\colon\thinspace{\mathcal{D}}^{(p)}\to M$ be the maximal
integral curve with starting point $p$.

  3. (3)

The flow of $X$ is the map $\theta\colon\thinspace{\mathcal{D}}\to M$ given by
$\theta_{t}(p)=\gamma_{p}(t)$ for $(t,p)$ in the domain
${\mathcal{D}}\coloneqq\\{({\mathcal{D}}^{(p)},p)\mid p\in M\\}$.

  4. (4)

$X$ is called complete if the domain of its flow is ${\bf R}\times M$.

It follows from the theory of ordinary differential equations, that for any
starting point integral curves exist and are uniquely determined by vector
fields. For details see [46, Chapter 17]. With some more work one can show the
existence and uniqueness of the flow of $X$ [46, Theorem 17.8].

######  Theorem 2.4.4 (Fundamental Theorem on Flows).

Let $X$ be a smooth vector field on a smooth manifold $M$. The flow
$\theta\colon\thinspace{\mathcal{D}}\to M$ of $X$ exists, is unique and has
the following properties:

  1. (1)

For each $p\in M$, the curve
$\theta^{(p)}\colon\thinspace{\mathcal{D}}^{(p)}\to M$ is the unique maximal
integral curve starting at $p$.

  2. (2)

If $s\in{\mathcal{D}}^{(p)}$, then ${\mathcal{D}}^{(\theta(s,p))}$ is the
interval ${\mathcal{D}}^{p}-s=\\{t-s\mid t\in{\mathcal{D}}^{(p)}\\}$.

  3. (3)

For each $t\in{\bf R}$, the set $M_{t}=\\{p\in M\mid(t,p)\in{\mathcal{D}}\\}$
is open in $M$, and $\theta_{t}\colon\thinspace M_{t}\to M_{-t}$ is a
diffeomorphism with inverse $\theta_{-t}$.

  4. (4)

For each $(t,p)\in{\mathcal{D}}$, $(d\theta_{t})_{p}X_{p}=X_{\theta_{t}(p)}$.

In order to get acquainted with these concepts, here are a few exercises.

######  Exercise 2.4.5.

Consider the vector field $X$ on ${\bf R}^{2}$ given by

$X(x,y)=\frac{\partial}{\partial x}+(1+y^{2})\frac{\partial}{\partial x}.$

Compute the flow of $X$ and its domain. Sketch its flow lines.

######  Exercise 2.4.6.

Show that

$X(x,y,z)=(-y,x,0)$

determines a smooth vector field on $S^{2}\subset{\bf R}^{3}$. What are the
integral curves of $X$ and what is the flow of $X$?

######  Exercise 2.4.7.

Let $M$ be a smooth manifold, $g$ a Riemannian metric on $M$ and
$f\colon\thinspace M\to{\bf R}$ a differentiable function. Consider the
gradient vector field $\operatorname{grad}f\in C^{\infty}(TM)$. Let
$\gamma\colon\thinspace(-\varepsilon,\varepsilon)\to M$ an integral curve of
$\operatorname{grad}f$. Show that $f\circ\gamma$ is monotonically increasing.

######  Lemma 2.4.8.

Suppose $f\colon\thinspace M\to N$ is a smooth map, $X$ and $Y$ are vector
fields on $M$ and $N$ respectively and $\theta$ and $\eta$ flows of $X$ and
$Y$ respectively. Then $X$ and $Y$ are $f$–related if and only if
$\eta_{t}\circ f(p)=f\circ\theta_{t}(p)$ for all $(t,p)$ in the domain
${\mathcal{D}}$ of $\theta$:

(2.4.1) $\begin{diagram}$

###### Proof.

The commutativity of Diagram (2.4.1) is equivalent to

$\eta^{f(p)}(t)=f\circ\theta^{(p)}(t)\quad\text{for all
}(t,p)\in{\mathcal{D}}.$

If $X$ and $Y$ are $f$–related, then the curvce
$\gamma\colon\thinspace{\mathcal{D}}^{(p)}\to N$ given by

$\gamma=f\circ\theta^{(p)}$

satisfies

$\gamma^{\prime}(t)=(f\circ\theta^{(p)})^{\prime}(t)=df(\theta^{(p)}{}^{\prime}(t))=df(X_{\theta^{(p)}(t)})=Y_{f\circ\theta^{(p)}(t)}=Y_{\gamma(t)}\quad\text{for
all }t\in{\mathcal{D}}^{(p)}$

Therefore $\gamma$ is an integral curve with starting point
$f\circ\theta^{(p)}(0)=f(p)$. By uniqueness of integral curves, the maximal
integral curve $\eta^{f(p)}$ must be defined on ${\mathcal{D}}^{(p)}$ and
$\gamma=\eta^{f(p)}$. Therefore Diagram (2.4.1) commutes.

Conversely, if Diagram (2.4.1) commutes, then $X$ and $Y$ are $f$–related by

$df_{p}(X_{p})=df_{p}(\theta^{(p)}{}^{\prime}(0))=(f\circ\theta^{(p)})^{\prime}(0)=\eta^{(f(p))}{}^{\prime}(0)=Y_{f(p)}\quad\text{for
all }p\in M.\qed$

The proof of the following exercise is in the spirit of Lemma 2.4.8.

######  Exercise 2.4.9.

Let $X$ and $Y$ be smooth vector fields on $M$ and $\theta$ and $\eta$ the
flows of $X$ and $Y$ respectively. Then $[X,Y]$ is equivalent to $X$ being
invariant under $\eta$.

######  Theorem 2.4.10 (Frobenius’ Theorem).

The following are equivalent:

  1. (1)

There exists a flat chart for ${\mathcal{D}}$ at every point of $M$;

  2. (2)

${\mathcal{D}}$ is completely integrable;

  3. (3)

${\mathcal{D}}$ is involutive.

###### Proof.

Certainly for any $p\in M$, if there exists a flat chart for ${\mathcal{D}}$,
and the slice of the form $x^{k+1}=p^{k+1},\ldots,x^{n}=p^{n}$ is an integral
manifold through $p$. Therefore existence of a flat chart at every point
implies complete integrability.

To see that complete integrability implies involutivity, consider two smooth
vector fields $X$ and $Y$ lying in a completely integrable distribution
${\mathcal{D}}$ on $M$ and $p\in M$. Let $(H,i)$ be an integral manifold of
${\mathcal{D}}$ through $p$ and suppose $i(q)=p$. Since $di\colon\thinspace
TN\to{\mathcal{D}}|_{i(N)}$ is a bundle isomorphism, there exist smooth vector
fields $\tilde{X}$ and $\tilde{Y}$ on $N$, such that $di\circ\tilde{X}=X\circ
i$ and $di\circ\tilde{Y}=Y\circ i$. By Exercise 2.2.4 $[X,Y]$ and
$[\tilde{X},\tilde{Y}]$ are $i$–related. Therefore
$[X,Y]_{p}=di_{q}([\tilde{X},\tilde{Y}]_{q})\in{\mathcal{D}}_{p}$, and
${\mathcal{D}}$ is involutive.

Now assume that ${\mathcal{D}}$ is a $k$–dimensional involutive distribution,
$\dim M=n$ and $p\in M$. Since the existence of a flat chart at $p$ is a local
question, we may replace $M$ by an open set $U\subset{\bf R}^{n}$ with $0=p\in
U$ and assume

${\mathcal{D}}_{p}\oplus\operatorname{span}\left\\{\left.\tfrac{\partial}{\partial
x^{k+1}}\right|_{p},\ldots\left.\tfrac{\partial}{\partial
x^{n}}\right|_{p}\right\\}=T_{p}{\bf R}^{n}.$

Choose a smooth local frame $Y_{1},\ldots,Y_{k}$ of ${\mathcal{D}}$ in $U$.
Let $\pi\colon\thinspace{\bf R}^{n}\to{\bf R}^{k}$ be the projection onto the
first $k$ coordinates. This induces a smooth bundle map $d\pi\colon\thinspace
TU\to T{\bf R}^{k}$ with

$d\pi\left(\sum_{i=1}^{n}a^{i}\left.\frac{\partial}{\partial
x^{i}}\right|_{q}\right)=\sum_{i=1}^{k}a^{i}\left.\frac{\partial}{\partial
x^{i}}\right|_{\pi(q)}$

By our choice of coordinates $\left.d\pi_{p}\right|_{{\mathcal{D}}_{p}}$ is
bijective, by continuity $\left.d\pi\right|_{{\mathcal{D}}_{q}}$ is
bijective333Consider e.g. the determinant map on the matrix with respect to a
local frame. for $q$ in an open neighborhood $U^{\prime}$ of $p$. Furthermore,
$\left.d\pi\right|_{{\mathcal{D}}}$ is a smooth bundle map and therefore the
matrix entries of $\left.d\pi\right|_{{\mathcal{D}}}$ with respect to our
local frame $(Y_{i})$ are smooth functions of $q\in U^{\prime}$, thus so are
the matrix entries of
$\left(\left.d\pi\right|_{{\mathcal{D}}_{q}}\right)^{-1}\colon\thinspace T{\bf
R}^{k}_{\pi(q)}\to{\mathcal{D}}_{q}$. Define a smooth local frame
$X_{1},\ldots,X_{k}$ for ${\mathcal{D}}$ in $U^{\prime}$ by

$\left.X_{i}\right|_{q}=\left(\left.d\pi_{q}\right|_{{\mathcal{D}}_{q}}\right)^{-1}\left.\frac{\partial}{\partial
x^{i}}\right|_{\pi(q)}.$

The vector fields $X_{i}$ and $\left.\tfrac{\partial}{\partial
x^{i}}\right|_{\pi(q)}$ are $\pi$–related since

$\left.\frac{\partial}{\partial
x^{i}}\right|_{\pi(q)}=\left(\left.d\pi_{q}\right|_{{\mathcal{D}}_{q}}\right)\left.X_{i}\right|_{q}=d\pi_{q}\left.X_{i}\right|_{q}.$

Therefore by Exercise 2.2.4

$d\pi([X_{i},X_{j}]_{q})=\left[\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial x^{j}}\right]_{\pi(q)}=0.$

Since $[X_{i},X_{j}]_{q}\in{\mathcal{D}}_{q}$ by involutivity and $d\pi$ is
injective on each fiber of ${\mathcal{D}}$, $[X_{i},X_{j}]_{q}=0$.

Now let $\theta_{i}$ be the flow of $X_{i}$ for $i=1,\ldots,k$. There exists
an $\varepsilon>0$ and a neighborhood $W$ of $p$ such that the composition
$(\theta_{1})_{t_{1}}\circ\ldots\circ(\theta_{k})_{t_{k}}$ is defined on $W$
and maps $W$ into $U^{\prime}$ whenever $|t_{1}|,\ldots,|t_{k}|$ are less than
$\varepsilon$. Let

$S\coloneqq\\{(u^{k+1},\ldots,u^{n})\mid(0,\ldots,0,u^{k+1},\ldots,u^{n})\in
W\\}$

and define $\psi\colon\thinspace(-\varepsilon,\varepsilon)^{k}\times S\to U$
by

$\psi(u^{1},\ldots,u^{n})\coloneqq(\theta_{1})_{u^{1}}\circ\ldots\circ(\theta_{k})_{u^{k}}(0,\ldots,0,u^{k+1},\ldots,u^{n})$

By Exercise 2.4.9 $X_{i}$ is invariant under the flow of $X_{j}$, in other
words $X_{i}$ is $(\theta_{j})_{s}$–related to itself. Therefore, since by
Lemma 2.4.8 all the flows $\theta_{i}$ commute with each other, for
$i\in\\{1,\ldots,k\\}$ and any $u_{0}\in W$ we have

$\displaystyle\left(d\psi_{u_{0}}\left.\frac{\partial}{\partial
u^{i}}\right|_{u_{0}}\right)f$ $\displaystyle=\left.\frac{\partial}{\partial
u^{i}}\right|_{u_{0}}f(\psi(u^{1},\ldots,u^{n}))$
$\displaystyle=\left.\frac{\partial}{\partial
u^{i}}\right|_{u_{0}}f((\theta_{1})_{u^{1}}\circ\ldots\circ(\theta_{k})_{u^{k}}(0,\ldots,0,u^{k+1},\ldots,u^{n}))$
$\displaystyle=\left.\frac{\partial}{\partial
u^{i}}\right|_{u_{0}}f((\theta_{i})_{u^{i}}\circ(\theta_{1})_{u^{1}}\circ\ldots\circ(\theta_{i-1})_{u^{i-1}}\circ(\theta_{i+1})_{u^{i+1}}\ldots\circ(\theta_{k})_{u^{k}}(0,\ldots,0,u^{k+1},\ldots,u^{n}))$
$\displaystyle=df\left(\left.\frac{\partial}{\partial
u^{i}}\right|_{u_{0}}(\theta_{i})_{u^{i}}\right)((\theta_{1})_{u^{1}}\circ\ldots\circ(\theta_{i-1})_{u^{i-1}}\circ(\theta_{i+1})_{u^{i+1}}\ldots\circ(\theta_{k})_{u^{k}}(0,\ldots,0,u^{k+1},\ldots,u^{n}))$
$\displaystyle=df(\left.X_{i}\right|_{\psi(u_{0})})=\left.X_{i}\right|_{\psi(u_{0})}(f)$

Then for $i=1,\ldots,k$ we have

$d\psi\left.\frac{\partial}{\partial u^{i}}\right|_{0}=(X_{i})_{p}.$

On the other hand, since

$\psi(0,\ldots,0,u^{k+1},\ldots,u^{n})=(0,\ldots,0,u^{k+1},\ldots,u^{n}),$

we get

$d\psi\left.\frac{\partial}{\partial
u^{i}}\right|_{0}=\left.\frac{\partial}{\partial
x^{i}}\right|_{p}\quad\text{for }i=k+1,\ldots,n.$

Therefore, $d\psi$ takes $(\left.\tfrac{\partial}{\partial
u^{1}}\right|_{0},\ldots,\left.\tfrac{\partial}{\partial u^{n}}\right|_{0})$
to $(X_{1}|_{p},\ldots,X_{k}|_{p},\left.\tfrac{\partial}{\partial
x^{1}}\right|_{p},\ldots,\left.\tfrac{\partial}{\partial x^{n}}\right|_{p})$,
which is also a basis. By the inverse function theorem, $\psi$ is a
diffeomorphism in a neighborhood of 0, and $\varphi=\psi^{-1}$ the flat
choordinate chart. ∎

######  Proposition 2.4.11.

Suppose that $f\colon\thinspace M\to N$ is $C^{\infty}$, that $(H,i)$ is a
$d$-dimensional integral manifold of a $k$–dimensional involutive distribution
${\mathcal{D}}$ on $N$, and that $f$ factors through $(H,i)$, that is,
$f(M)\subset i(H)$. Let $f_{0}\colon\thinspace M\to H$ be the (unique) mapping
such that $i\circ f_{0}=f$.

$\begin{diagram}$

Then $f_{0}$ is $C^{\infty}$.

###### Proof.

By the injectivity of $i$ there is a unique $q\in H$ for all $p\in M$ such
that $i(q)=f(p)$, i.e. there is a unique map $f_{0}$ such $i\circ f_{0}=f$.
Let $p\in M$ be arbitrary and set $q=f_{0}(p)\in H$. Let
$(y^{1},\ldots,y^{n})$ be flat coordinates for ${\mathcal{D}}$ on a
neighborhood $U$ of $i(q)$ and let $\pi\colon\thinspace U\to{\bf R}^{n-k}$ be
given by $\pi=(y^{k+1},\ldots,y^{n})$. Since $i$ is continuous, $i^{-1}(U)$ is
open in $H$ and therefore consists of a countable number of connected
components, each of which is open in $H$. Because ${\mathcal{D}}$ is spanned
by the first $k$ coordinate vector fields in $U$, this implies that
$\pi\circ\gamma$ is constant for any smooth path $\gamma$ in a connected
component of $i(i^{-1}(U))=i(H)\cap U$. Therefore each connected component of
$i(H)\cap U$ is a subset of a slice and $\pi(i(H)\cap U)$ consists of a
countable number of points. Choose smooth coordinates $(x^{i})$ for $M$ on a
connected neighborhood $V$ of $p$ such that $f(V)\subset U$ and $i^{-1}$.
Since $f(V)\subset i(H)\cap U$, $\pi\circ f(V)$ is also countable. Since $V$
is connected, the coordinate functions are actually constant, and thus $f(V)$
lies in a single slice. On this slice $(y^{1},\ldots,y^{k})$ are smooth
coordinates, so $f_{0}$ has the local coordinate representation
$(y^{1},\ldots,y^{k})\circ f(x)=(f^{1}_{0}(x),\ldots,f^{k}_{0}(x))$, which is
smooth. ∎

######  Theorem 2.4.12.

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let
$\tilde{\mathfrak{h}}\subset\mathfrak{g}$ be a subalgebra. Then there is a
unique connected Lie subgroup $(H,\varphi)$ of $G$ such that
$d\varphi(\mathfrak{h})=\tilde{\mathfrak{h}}$.

###### Proof.

Let $d\coloneqq\dim\tilde{\mathfrak{h}}$. Consider the smooth $d$–dimensional
distribution ${\mathcal{D}}$ given by

${\mathcal{D}}(g)\coloneqq\\{X(g)\mid
X\in\tilde{\mathfrak{h}}\\}\quad\text{for every }g\in G.$

${\mathcal{D}}$ is globally spanned by a basis $X_{1},\ldots,X_{d}$ for
$\tilde{\mathfrak{h}}$. ${\mathcal{D}}$ is involutive, for if $X$ and $Y$ are
vector fields lying in ${\mathcal{D}}$, then there are $C^{\infty}$ functions
$\\{a_{i}\\}$ and $\\{b_{i}\\}$ on $G$ such that $X=\sum a_{i}X_{i}$ and
$Y=\sum b_{i}X_{i}$, and therefore

$[X,Y]=\sum_{i,j=1}^{d}\\{a_{i}b_{j}[X_{i},X_{j}]+a_{i}X_{i}(b_{i})Y_{j}-b_{j}Y_{j}(a_{i})X_{i}\\},$

which is a vector field in ${\mathcal{D}}$ since $\tilde{\mathfrak{h}}$ is a
subalgebra of $\mathfrak{g}$.

Let $(H,\varphi)$ be a maximal connected integral manifold of ${\mathcal{D}}$
through $e$. Let $g\in\varphi(H)$. Since ${\mathcal{D}}$ is invariant under
left translations, $(H,l_{g^{-1}}\circ\varphi)$ is also an integral manifold
of ${\mathcal{D}}$ through $e$ with
$l_{g^{-1}}\circ\varphi(H)\subset\varphi(H)$. Therefore, $g,h\in\varphi(H)$
implies $g^{-1}h\in\varphi(H)$ and $\varphi(H)$ is a subgroup of $G$. Equip
$H$ with the group structure so that $\varphi$ is a homomorphism.

We need to check that $H$ is a Lie group, i.e. that $\alpha:H\times H\to H$
with $\alpha(g,h)=gh^{-1}$ is $C^{\infty}$. Since $H$ is a subgroup of $G$,
the map $\beta\colon\thinspace H\times H\to G$ given by
$\beta(g,h)=\varphi(g)\varphi(h)^{-1}$ is $C^{\infty}$ and the following
diagram commutes:

$\begin{diagram}$

Since $H$ is an integral manifold of an involutive distribution on $G$,
$\alpha$ is also $C^{\infty}$ by Proposition 2.4.11. Thus $(H,\varphi)$ is a
Lie subgroup of $G$, and if $\mathfrak{h}$ is the Lie algebra of $H$,
$d\varphi(\mathfrak{h})=\tilde{\mathfrak{h}}$ for dimension reasons by Theorem
2.3.2.

To prove uniqueness, let $(K,\psi)$ be another connected Lie subgroup of $G$
with $d\varphi(\mathfrak{k})=\tilde{\mathfrak{h}}$. Then $(K,\varphi)$ is also
an integral manifold of ${\mathcal{D}}$ through $e$. By the maximality of
$(H,\varphi)$, $\psi(K)\subset\varphi(H)$, and there is a uniquely determined
map $\psi_{0}$ such that $\varphi\circ\psi_{0}=\psi$

$\begin{diagram}$

By Proposition 2.4.11 $\psi_{0}$ is smooth, and therefore is an injective Lie
group homomorphism. Also, $\psi_{0}$ is everywhere non-singular, in particular
it is a diffeomorphism on a neighborhood of the identity, and therefore
surjective by Proposition 2.4.2. Thus $\psi_{0}$ is a Lie group isomorphism.
This proves uniqueness. ∎

######  Corollary 2.4.13.

There is a 1:1 correspondence between connected Lie subgroups of a Lie group
and subalgebras of its Lie algebra.

Recall that a topological space is simply connected, if its fundamental group
is trivial. Then one can show the following theorem with some more work on
covering spaces. For a proof see [62, Theorem 3.28].

######  Theorem 2.4.14.

There is a 1:1 correspondence between isomorphism classes of Lie algebras and
isomorphism lasses of simply connected Lie groups.

###  2.5. The exponential map

Let $G$ be a Lie group, and let $\mathfrak{g}$ be its Lie algebra.

######  Definition 2.5.1.

A homomorphism $\varphi\colon\thinspace{\bf R}\to G$ is called a 1–parameter
subgroup of $G$.

We will see that 1–parameter subgroups are precisely the integral curves of
left-invariant vector fields.

######  Lemma 2.5.2.

Every left-invariant vector field on $G$ is complete.

###### Proof.

Let $G$ be a Lie group, let $X\in\mathfrak{g}$, and let $\theta$ denote the
flow of $X$. Suppose some maximal integral curve $\theta^{(g)}$ with starting
point $g$ is defined on an interval $(a,b)\subset{\bf R}$, and assume
$b<\infty$. (The case $a>-\infty$ is handled analogously.) By Lemma 2.4.8

(2.5.1) $l_{g}\circ\theta_{t}(h)=\theta_{t}\circ l_{g}(h)$

on the domain of $\theta$. The integral curve $\theta^{(e)}$ starting at the
identity is at least defined on $(-\varepsilon,\varepsilon)$ for some
$\varepsilon>0$. Choose $c\in(b-\varepsilon,b)$ and define the curve
$\gamma\colon\thinspace(a,c+\varepsilon)\to G$ by

$\gamma(t)=\begin{cases}\theta^{(g)}(t)&t\in(a,b)\\\
l_{\theta^{(g)}(c)}(\theta^{(e)}(t-c))&t\in(c-\varepsilon,c+\varepsilon).\end{cases}$

By (2.5.1) we have for $t\in(a,b)\cap(c-\varepsilon,c+\varepsilon)$

$\displaystyle l_{\theta^{(g)}(c)}(\theta^{(e)}(t-c))$
$\displaystyle=l_{\theta^{(g)}(c)}(\theta_{t-c}(e))$
$\displaystyle=\theta_{t-c}(l_{\theta^{(g)}(c)}(e))=\theta_{t-c}(\theta_{c}(g))$
$\displaystyle=\theta_{t}(g)=\theta^{(g)}(t),$

where $\theta_{t-c}\circ\theta_{c}=\theta_{t}$ by the Fundamental Theorem of
Flows, so the definitions for $\gamma$ agree on the overlap.

On $(a,b)$ is an integral curve of $X$, for
$t_{0}\in(c-\varepsilon,t+\varepsilon)$ we compute using the chain-rule and
the left-invariance of $X$

$\displaystyle\gamma^{\prime}(t_{0})$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=t_{0}}l_{\theta^{(g)}(c)}(\theta^{(e)}(t-c))=dl_{\theta^{(g)}(c)}\left(\left.\frac{d}{dt}\right|_{t=t_{0}}\theta^{(e)}(t-c)\right)$
$\displaystyle=dl_{\theta^{(g)}(c)}(X_{\theta^{(e)}(t_{0}-c)})=X_{\gamma(t_{0})}.$

Therefore $(a,b)$ is not the maxmial of $\theta^{(g)}$. ∎

Using (2.5.1) again, we can easily show the following.

######  Exercise 2.5.3.

Let $X\in\mathfrak{g}$. The integral curve of $X$ starting at $e$ is a one-
parameter subgroup of $G$.

######  Theorem 2.5.4.

Every one-parameter subgroup of $G$ is an integral curve of a left-invariant
vector field. Thus there are one-to-one correspondences

$\\{\text{one-parameter subgroups of
}G\\}\longleftrightarrow\mathfrak{g}\longleftrightarrow T_{e}G$

In particular, a one-parameter subgroup is uniquely determined by its initial
tangent vector in $T_{e}G$.

###### Proof.

Let $\varphi\colon\thinspace{\bf R}\to G$ be a one-parameter subgroup, and let
$X=d\varphi\left(\frac{d}{dt}\right)\in\mathfrak{g}$, where $\frac{d}{dt}$ is
considered as a left-invariant vector field on ${\bf R}$. Since $X$ is
$\varphi$–related to $\frac{d}{dt}$ we have

$\varphi^{\prime}(t_{0})=d\varphi_{t_{0}}\left.\left(\frac{d}{dt}\right|_{t_{0}}\right)=X_{t_{0}}.$

Therefore $\varphi$ is the integral curve of $X$. Together with Exercise 2.5.3
and Theorem 2.3.2 the proof is complete. ∎

######  Definition 2.5.5.

We call the integral curve $\varphi$ of $X$ the one-parameter subgroup
generated by $X$. The exponential map of $G$ is given by

$\exp(X)\coloneqq\varphi(1).$

The following proposition explains the choice of name.

######  Proposition 2.5.6.

For any matrix $A\in\mathfrak{gl}(n,{\bf R})$, let

$e^{A}=\sum_{k=0}^{\infty}\frac{1}{k!}A^{k}.$

This series converges to an invertible matrix, and the one-parameter subgroup
of $\text{\it{GL}}(n,{\bf R})$ generated by the left-invariant vector field
$\tilde{A}$ corresponding to $A$ is $\varphi(t)=e^{tA}$.

###### Proof.

Since $\|AB\|\leq\|A\|\|B\|$ and therefore $\|A^{k}\|\leq\|A\|^{k}$ with the
standard norm on ${\bf R}^{n^{2}}$, the series converges uniformly and
$\|e^{A}\|\leq e^{\|A\|}$.

After differentiating each summand, the series still converges uniformly on
bounded sets. Therefore term-by-term differentiation is justified and
$\varphi(t)=e^{tA}$ satisfies the differential equation
$\varphi^{\prime}(t)=\varphi(t)A$ as well as $\varphi(0)=e$.

Consider the left-invariant vector field $\tilde{A}$ corresponding to $A$.
Therefore, we have for our (global) coordinate functions $x_{ij}$ on
$\text{\it{GL}}(n,{\bf R})$ that $\tilde{A}_{e}(x_{ij})=A_{ij}$. The one-
parameter subgroup $\gamma$ generated by $\tilde{A}$ is the unique integral
curve of $\tilde{A}$ starting at the identity, i.e. it is the unique solution
to the initial value problem

$\displaystyle\gamma^{\prime}(t)$
$\displaystyle=\tilde{A}_{\gamma(t)}=dl_{\gamma(t)}\tilde{A}_{e}$
$\displaystyle\gamma(0)$ $\displaystyle=e,$

which written in matrix form via our coordinates $x_{ij}$ can be easily seen
to be equivalent to

$\displaystyle\gamma^{\prime}(t)$ $\displaystyle=\gamma(t)A$
$\displaystyle\gamma(0)$ $\displaystyle=e.$

Therefore $\gamma(t)=\varphi(t)$. ∎

The following proposition is not surprising, but is not apparent in our
previous discussion. In particular, it tells us what the one-parameter
subgroups of matrix Lie groups are.

######  Proposition 2.5.7.

Let $H\subset G$ be a Lie subgroup. The one-parameter subgroups of $H$ are
precisely those one-parameter subgroups of $G$ whose initial tangent vectors
lie in $T_{e}H$.

###### Proof.

Let $\varphi\colon\thinspace{\bf R}\to H$ be a one-parameter subgroup. Then

${\bf R}\stackrel{{\scriptstyle\varphi}}{{\longrightarrow}}H\hookrightarrow G$

is a Lie group homomorphism and thus a one-parameter subgroup of $G$.

On the other hand, if $\varphi\colon\thinspace{\bf R}\to G$ and
$\psi\colon\thinspace{\bf R}\to H$ are one-parameter subgroup with the same
initial vector in $T_{e}H$, then

${\bf R}\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}H\hookrightarrow G$

must be equal $\varphi$. ∎

######  Proposition 2.5.8 (Properties of the exponential map).

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra.

  1. (1)

The exponential map $\exp\colon\thinspace\mathfrak{g}\to G$ is smooth.

  2. (2)

For any $X\in\mathfrak{g}$, $\varphi(t)=\exp(tX)$ is the one-parameter
subgroup of $G$ generated by $X$.

  3. (3)

For any $X\in\mathfrak{g}$, $\exp(s+t)X=\exp(sX)\exp(tX)$.

  4. (4)

Under the canonical identifications of $T_{0}\mathfrak{g}$ and $T_{e}G$,
$d\exp_{0}={\rm id}$.

  5. (5)

The exponential map restricts to a diffeomorphism from a neighborhood of 0 in
$\mathfrak{g}$ to a neighborhood of $e$ in $G$.

  6. (6)

If $H$ is another Lie group and $\mathfrak{h}$ its Lie algebra, then for any
Lie group homomorphism $f\colon\thinspace G\to H$ the following diagram
commutes:

$\begin{diagram}$

  7. (7)

The flow $\theta$ of $X\in\mathfrak{g}$ is given by $\theta_{t}=r_{\exp(tX)}$,
where $r$ denotes right translation.

###### Proof.

For any $X\in\mathfrak{g}$ let $\theta_{(X)}$ its flow.

  1. (1)

Define a smooth vector field $Y$ on $G\times\mathfrak{g}$ by

$Y_{(g,X)}\coloneqq(X_{g},0)\in T_{g}G\times T_{X}\mathfrak{g}\cong
T_{(g,X)}(G\times\mathfrak{g})$

The flow of $Y$ is given by $\eta_{t}(g,X)\coloneqq(\theta_{(X)}(t,g),X)$. By
the fundamental theorem of flows, Theorem 2.4.4, $\eta$ is smooth. Since $\exp
X=\pi_{1}(\theta_{1}(e,X))$, where $\pi_{1}\colon\thinspace
G\times\mathfrak{g}\to G$ is the projection, $\exp$ is smooth.

  2. (2)

We need to show that $\exp(tX)=\theta_{(X)}^{(e)}(t)$, in other words

$\theta_{(tX)}^{(e)}(1)=\theta_{(X)}^{(e)}(t).$

Fix $t\in{\bf R}$ and define $\gamma(s)=\theta_{(X)}^{(e)}(st)$. Then by the
chain rule,

$\gamma^{\prime}(s)=t(\theta_{(X)}^{(e)})^{\prime}(st)=tX_{\gamma(s)}.$

So $\gamma$ is integral curve of $tX$. Since $\gamma(0)=e$, by the uniqueness
of integral curves we have $\gamma(s)=\theta^{(e)}_{(tX)}(s)$. For $s=1$ we
get the desired formula.

  3. (3)

This follows from the one-parameter group being a group homomorphism.

  4. (4)

Let $X\in T_{e}G=T_{0}\mathfrak{g}$ be arbitrary and $\sigma(t)=tX$. Then
$\sigma^{\prime}(0)=X$ and by (2) we have

$d\exp_{0}X=d\exp_{0}\sigma^{\prime}(0)=(\exp\circ\sigma)^{\prime}(0)=\left.\frac{d}{dt}\right|_{t=0}\exp
tX=X.$

  5. (5)

This follows from (4) by the inverse function theorem.

  6. (6)

By (2) $\exp(tdf(X))$ is the one-parameter subgroup generated by $df(X)$. Let
$\sigma(t)\coloneqq f(\exp(tX))$. Then

$\sigma^{\prime}(0)=\left.\frac{d}{dt}\right|_{t=0}f(\exp(tX))=df\left(\left.\frac{d}{dt}\right|_{t=0}\exp(tX)\right)=df(X)$

and

$\sigma(s+t)=f(\exp(s+t)X)\stackrel{{\scriptstyle(3)}}{{=}}f(\exp(sX)\exp(tX))=f(\exp(sX))f(\exp(tX))=\sigma(s)\sigma(t).$

So $\sigma$ is also a one-parameter subgroup satisying the same differential
equation and therefore equals $\exp(tdf(X))$.

  7. (7)

By (2) and Lemma 2.4.8 we have

$r_{\exp(tX)}(g)=g\exp(tX)=l_{g}(\exp(tX))=l_{g}(\theta_{(X)t}(e))=\theta_{(X)t}(l_{g}(e))=\theta_{(X)t}(g).\qed$

###  2.6. Lie Group actions

Let $G$ be a Lie group and $M$ a smooth manifold.

######  Definition 2.6.1.

A left action of $G$ on $M$ is a smooth map

$\displaystyle\theta\colon\thinspace G\times M$ $\displaystyle\to M$
$\displaystyle(g,p)$ $\displaystyle\mapsto\theta_{g}(p)=g\cdot p$

such that

$g\cdot(h\cdot p)=(g\cdot h)\cdot p\quad\text{and}\quad e\cdot
p=p\quad\text{for all }g,h\in G,p\in M.$

A right action of $G$ on $M$ is a smooth map

$\displaystyle M\times G$ $\displaystyle\to M$ $\displaystyle(p,g)$
$\displaystyle\mapsto p\cdot g$

such that

$(p\cdot g)\cdot h=g\cdot(g\cdot h)\quad\text{and}\quad p\cdot
e=p\quad\text{for all }g,h\in G,p\in M.$

We will list some frequently used terminology regarding group actions.
Consider a left action of $G$ on $M$.

  * •

The orbit of $p$ under the action is the set $G\cdot p\coloneqq\\{g\cdot p\mid
g\in G\\}$.

  * •

The isotropy group of $p$ is given by $G_{p}\coloneqq\\{g\mid g\cdot p=p\\}$.

  * •

The action is transitive if for any $p,q\in M$ there is a $g\in G$ such that
$g\cdot p=q$.

  * •

The action is free if $G_{p}=\\{e\\}$ for all $p\in M$.

  * •

$p\in M$ is a fixed point of the action, if $g\cdot p=p$ for all $g\in G$.

A few relevant examples are the following:

  1. (1)

Any Lie group acts freely and transivitely on itself by multiplication.

  2. (2)

Any Lie group $G$ also acts on itself by conjugation $C_{g}h\coloneqq
ghg^{-1}$, for which $G_{h}$ are all elements which commute with $g$. Notice
that $C_{g}$ is a Lie group homomorphism.

  3. (3)

$O(n)$ naturally acts on ${\bf R}^{n}$ and restricts to a transitive action on
$S^{n-1}$.

Using the following concept we can generate a lot more important examples.

######  Definition 2.6.2.

If $G$ is a Lie group, a (finite-dimensional) representation of $G$ is a Lie
group homomorphism $\rho\colon\thinspace G\to\text{\it{GL}}(V)$ for some
finite-dimensional real or complex vector space. We will also encounter
(possibly infinite-dimensional) representations on Hilbert spaces, where
$\text{\it{GL}}(V)$ gets replaced by the group of bounded linear operators on
a Hilbert space. If $\rho$ is injective, $\rho$ is called faithful.

Any representation $\rho$ yields a left action $g\cdot v\coloneqq\rho(g)v$ for
$g\in G,v\in V$. If $G$ is a Lie subgroup of $\text{\it{GL}}(n,{\bf R})$, then
$G\hookrightarrow\text{\it{GL}}(n,{\bf R})$ is a faithful representation,
called the defining representation.

If $G$ is a Lie group and $\mathfrak{g}$ its Lie algebra, consider the Lie
algebra homomorphism $\operatorname{Ad}$ induced by $C_{g}$. By the following
theorem $\operatorname{Ad}\colon\thinspace G\to\text{\it{GL}}(\mathfrak{g})$
and $\operatorname{Ad}$ is a representation of $G$, called the adjoint
representation.

######  Theorem 2.6.3.

Let $\theta\colon\thinspace G\times M\to M$ be a left action for which $p$ is
a fixed point, then the map

$\displaystyle\rho\colon\thinspace G$
$\displaystyle\to\operatorname{Aut}(T_{p}M)$ $\displaystyle g$
$\displaystyle\mapsto(d\theta_{g})_{p}$

is a representation of $G$.

###### Proof.

Notice that $\rho$ not only maps into the endomorphisms of $T_{p}M$ but the
Lie group of automorphisms, because $\rho(e)=\operatorname{Id}$ and

$\rho(gh)=(d\theta_{gh})p=d(\theta_{g}\circ\theta_{h})_{p}=\rho(g)\rho(h)$

imply that $\rho(g)$ is invertible with $\rho(g)^{-1}=\rho(g^{-1})$. The above
remarks also show, that $\rho$ is a homomorphism of groups.

It remains to show that $\rho$ is $C^{\infty}$. It suffices to show that
$\rho$ composed with any coordinate function on $\operatorname{Aut}(T_{p}M)$
is $C^{\infty}$. One gets a coordinate system by using a basis of $T_{p}M$
(and its dual basis) to identify $\operatorname{Aut}(T_{p}M)$ with non-
singular matrices: a coordinate function on $\operatorname{Aut}(T_{p}M)$ is of
the form

$\displaystyle\varphi\colon\thinspace\operatorname{Aut}(T_{p}M)$
$\displaystyle\to{\bf R}$ $\displaystyle A$ $\displaystyle\mapsto v^{*}(Av),$

where $v\in T_{p}M$ is a basis vector and $v^{*}$ is the dual vector with
respect to the basis. Then $\varphi\circ\rho$ is smooth, because it is the
composition of $C^{\infty}$ maps

$G\stackrel{{\scriptstyle i}}{{\to}}T(G)\times
T(M)\stackrel{{\scriptstyle\cong}}{{\to}}T(G\times M)\stackrel{{\scriptstyle
d\theta}}{{\to}}T(M)\stackrel{{\scriptstyle v^{*}}}{{\to}}{\bf R}$

where $i(g)=((g,0),(p,v))$. ∎

Recall that the equivalent of a directional derivative in Euclidean space is
the Lie derivative $L_{X}Y$ of smooth vector fields $X$ and $Y$ given by

$(L_{X}Y)_{p}=\left.\frac{d}{dt}\right|_{t=0}d(\theta_{-t})Y_{\theta_{t}(p)}=\lim_{t\to
0}\frac{d(\theta_{-t})Y_{\theta_{t}(p)}-Y_{p}}{t},$

where $\theta$ is the flow of $X$.

We will need to know the following important result.

######  Proposition 2.6.4.

For any smooth vector fields $X$ and $Y$ on a smooth manifold $M$

$L_{X}Y=[X,Y].$

###### Proof.

Let $f\in C^{\infty}(M)$. Let $\theta_{t}$ be the flow of $X$ and define
$F(t,p):=f(\theta_{t}(p))-f(p)$ and $F^{\prime}:=\tfrac{\partial}{\partial
t}F$. Set

$g_{t}(p):=\int_{0}^{1}F^{\prime}(ts,p)ds.$

Then $g\colon\thinspace(-\varepsilon,\varepsilon)\times M\to{\bf R}$ is smooth
and satisfies for $t>0$

$tg_{t}(p)=\int_{0}^{1}tF^{\prime}(ts,p)ds=\int_{0}^{t}F^{\prime}(\tilde{s},p)d\tilde{s}=F(t,p).$

Similarly we can extend this to $tg_{t}(p)=f(\theta_{t}(p))-f(p)$ for all
$t\in(-\varepsilon,\varepsilon)$, which yields

$X_{p}f=\left.\frac{d}{dt}\right|_{t=0}f(\theta_{t}(p))=\lim_{t\to
0}\frac{f(\theta_{t}(p))-f(p)}{t}=\lim_{t\to 0}g_{t}(p)=g_{0}(p)$

as well as

$(d\theta_{-t})Y_{\theta_{t}(p)}f=Y_{\theta_{t}(p)}(f\circ\theta_{-t})=Y_{\theta_{t}(p)}f-tY_{\theta_{t}(p)}(g_{-t}).$

Therefore,

$\displaystyle(L_{X}Y)_{p}(f)$ $\displaystyle=\lim_{t\to
0}\frac{d(\theta_{-t})Y_{\theta_{t}(p)}f-Y_{p}f}{t}=\lim_{t\to
0}\frac{Y_{\theta_{t}(p)}f-Y_{p}f}{t}-\lim_{t\to 0}Y_{\theta_{t}(p)}(g_{-t})$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0}Y(f\circ\theta_{t}(p))-Y_{p}g_{0}=X_{p}(Yf)-Y_{p}(Xf)=[X,Y]_{p}f.\qed$

A Lie algebra homomorphism $\mathfrak{g}\to End(V)$ for some vector space $V$
is also known as a representation of $\mathfrak{g}$.

######  Theorem 2.6.5.

Let $G$ be a Lie group, let $\mathfrak{g}$ be its Lie algebra. Then the Lie
algebra representation induced by $\operatorname{Ad}$

$\operatorname{ad}\colon\thinspace\mathfrak{g}\to\mathfrak{gl}(\mathfrak{g})$

is given by $\operatorname{ad}(X)Y=[X,Y]$.

###### Proof.

We have

$\operatorname{ad}(X)Y=\left(\left.\frac{d}{dt}\right|_{t=0}\operatorname{Ad}(\exp(tX))\right)Y=\left.\frac{d}{dt}\right|_{t=0}(\operatorname{Ad}_{\exp(tX)}(Y))=\left.\frac{d}{dt}\right|_{t=0}d\left(C_{\exp(tX)}\right)(Y).$

By Theorem 2.5.8 the flow of $X$ is given by $\theta_{t}=r_{exp(tX)}$. Then

$\displaystyle(\operatorname{ad}(X)Y)_{e}$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0}d(r_{\exp(-tX)})\left(d(l_{\exp(tX)})(Y)\right)$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0}d(r_{\exp(-tX)})\left(Y_{\exp(tX)}\right)$
$\displaystyle=\left.\frac{d}{dt}\right|_{t=0}d(\theta_{-t})\left(Y_{\theta_{t}(e)}\right)$
$\displaystyle=(L_{X}Y)(e)=[X,Y](e).$

Since left-invariant vector fields are determined by their value at $e$, this
completes the proof. ∎

###  2.7. Principal bundles

######  Definition 2.7.1.

Let $M$ be a manifold and $G$ be a Lie group. A principal $G$–bundle over $M$
consists of a smooth manifold $P$ and an action of $G$ on $P$ satisfying

  1. (1)

$G$ acts freely on the right;

  2. (2)

$M$ is the quotient space of $P$ by the equivalence relation induced by $G$,
$M=P/G$ and the canonical projection $\pi\colon\thinspace P\to M$ is
differentiable;

  3. (3)

$P$ is locally trivial, that is, every point $x\in M$ has a neighborhood $U$
such that there exists an diffeomorphism $\psi\colon\thinspace\pi^{-1}(U)\to
U\times G$ with

    1. (a)

$\psi$ is equivariant: $\psi(p\cdot g)=\psi(p)\cdot g$, where $(x,h)\cdot
g=(x,h\cdot g)$ for $x\in U$ and $g,h\in G$;

    2. (b)

$\pi(\psi^{-1}(x,g))=x$.

$P$ is called the total space, $M$ the base space, $G$ is the structure group,
$\pi$ the projection and $\pi^{-1}(x)$ the fiber over a point $x\in M$.

Given a principal bundle $P$, it is not difficult to choose an open covering
$\\{U_{\alpha}\\}$ of $M$ and with diffeomorphisms $\psi_{\alpha}$ from
Definition 2.7.1. Consider the transition functions $\psi_{\alpha\beta}$

$(U_{\alpha}\cap U_{\beta})\times
G\stackrel{{\scriptstyle\psi_{\alpha}^{-1}}}{{\longrightarrow}}\pi^{-1}(U_{\alpha}\cap
U_{\beta})\stackrel{{\scriptstyle\psi_{\beta}}}{{\longrightarrow}}(U_{\alpha}\cap
U_{\beta})\times G.$

Let $\pi_{2}\colon\thinspace U_{\alpha}\times G\to G$ be the projection to the
second coordinate. Then $\pi_{2}\circ\psi_{\alpha}(p\cdot
g)=\pi_{2}\circ\psi_{\alpha}(p)\cdot g$ and thus for
$p\in\pi^{-1}(U_{\alpha}\cap U_{\beta})$

$\pi_{2}\circ\psi_{\beta}(pg)(\pi_{2}\circ\psi_{\alpha}(pg))^{-1}=\pi_{2}\circ\psi_{\beta}(p)(\pi_{2}\circ\psi_{\alpha}(p))^{-1},$

which shows that $\psi_{\alpha\beta}$ only depends on $\pi(p)$ and not on $p$
and we have $\psi_{\alpha\beta}(x,g)=(x,gh_{\alpha\beta}(x))$ for some smooth
map $h_{\alpha\beta}\colon\thinspace U_{\alpha}\cap U_{\beta}\to G$.

If $s\colon\thinspace M\to P$ is a section of a principal $G$–bundle
$\pi\colon\thinspace P\to M$, i.e. $s$ is smooth with $\pi\circ s=\text{id}$,
then $s$ determines a trivialization $P\cong M\times G$ by
$p\mapsto(\pi(p),g_{p})$, where $g_{p}\in G$ is determined by $p\cdot
g_{p}=s(\pi(p))$. $P$ is called trivializable444We want to stress that there
is a difference between the trivial bundle $M\times G$ and the trivializable
bundle $P$..

The frame bundle $\text{\it{GL}}(TM)\to M$ is the principal
$\text{\it{GL}}(n,{\bf R})$–bundle whose fiber over a point $x\in M$ is the
collection of all bases for $T_{x}M$; this fiber is (not canonically)
isomorphic to $\text{\it{GL}}(n,{\bf R})$. For a given Riemannian metric on
$M$ we have the orthonormal frame bundle by considering all orthonormal
frames. This forms a sub-principal bundle of $\text{\it{GL}}(TM)$ with fiber
$O(n)\subset\text{\it{GL}}(n,{\bf R})$.

######  Exercise 2.7.2.

If $M$ is orientable, then $O(TM)$ is the union of two components. A choice of
orientation for $M$ is the same as a choice of a component of $O(TM)$. This
component is a principal $\text{\it{SO}}(n)$–bundle, the oriented orthonormal
frame bundle.

You can construct a principal $\text{\it{GL}}(n,{\bf R})$– (or
$\text{\it{GL}}(n,{\bf C})$–) bundle over $M$ from a vector bundle $E\to M$.
If $E$ is equipped with a fiberwise positive definite symmetric inner product,
then its structure group can be reduced to $O(n)$ or $U(n)$555In this case,
$E$ is called a $O(n)$– or $U(n)$–vector bundle. and one can construct a
principal $O(n)$– (or $U(n)$–) bundle from $E$.

######  Exercise 2.7.3.

The Hopf fibration is a principal $S^{1}$–bundle over $S^{2}$, where the
projection $\pi\colon\thinspace S^{3}\subset{\bf C}^{2}\to S^{2}\subset{\bf
C}^{2}\times{\bf R}={\bf R}^{3}$ is given by
$\pi(z_{0},z_{1})=(2z_{0}\bar{z}1,|z_{0}|^{2}-|z_{1}|^{2})$.

######  Definition 2.7.4.

Let $P\to M$ be a principal $G$–bundle and $\rho\colon\thinspace
G\to\text{\it{GL}}(V)$ a representation of $G$. Then define (the Borel
construction)

$P\times_{\rho}V$

to be the quotient space $P\times V/\sim$ where

$(p,v)\sim(pg,g^{-1}v).$

######  Exercise 2.7.5.

The function $P\times_{\rho}V\to M$ given by $(p,v)\mapsto\pi(p)$ is well-
defined and makes $P\times_{\rho}V$ into a vector bundle over $M$ with fiber
$V$, the vector bundle associated to $P$ by the representation $V$.

For the defining representation of $\text{\it{SO}}(n)$ the associated vector
bundle to a principal $\text{\it{SO}}(n)$–bundle is a vector bundle with fiber
${\bf R}^{n}$. Similarly we can take the defining representation of $U(n)$.

For the Adjoint representation $\operatorname{Ad}$ the associated vector
bundle is called the Adjoint bundle and is denoted by
$\operatorname{Ad}_{P}\to M$.

######  Exercise 2.7.6.

Show that the adjoint representation of $\text{\it{SO}}(3)$ is isomorphic to
its defining representation
$i\colon\thinspace\text{\it{SO}}(3)\hookrightarrow\text{\it{GL}}(3,{\bf R})$:
find an SO–equivariant vector space isomorphism $\varphi\colon\thinspace{\bf
R}^{3}\to\mathfrak{so}(3)$ (also known as an intertwining map), i.e. an
isomorphism satisfying $\operatorname{Ad}(g)\varphi(v)=\varphi(i(g)v)$ for all
$g\in\text{\it{SO}}(3)$ and $v\in{\bf R}^{3}$.

Since $\text{\it{SU}}(2)$ is the double cover of $\text{\it{SO}}(3)$, the
adjoint representation of $\text{\it{SU}}(2)$ is the same as the composition
of the 2–fold cover $\text{\it{SU}}(2)\to\text{\it{SO}}(3)$ with the defining
representation of $\text{\it{SO}}(3)$.

######  Exercise 2.7.7.

We can define a homomorphism
$[\cdot,\cdot]\colon\thinspace\operatorname{Ad}P\otimes\operatorname{Ad}P\to\operatorname{Ad}P$
by setting $[(p,v),(p,v^{\prime})]:=(p,[v,v^{\prime}])$ on representatives
$(p,v)$ and $(p,v^{\prime})$ of elements in the fiber of $\pi(p)$. Check that
this homomorphism is well-defined.

A useful observation is that if $E=P\times_{\rho}V$, then sections of $E$ are
just $G$–equivariant maps $f\colon\thinspace P\to V$. The correspondence is
given by $(f\colon\thinspace P\to V)\mapsto(\varphi_{f}\in\Gamma(E))$ where
$\varphi_{f}(x)$ is the equivalence class of $(p,f(p))$ where
$p\in\pi^{-1}(x)$.

######  Exercise 2.7.8.

Check that this map makes sense and is a vector space homomorphism.

###  2.8. Homology with local coefficients

######  Definition 2.8.1.

The group ring ${\bf Z}\pi$ is a ring associated to a group $\pi$. Additively
it is the free abelian group on $\pi$ with multiplication given by

$\left(\sum_{i}m_{i}g_{i}\right)\left(\sum_{j}n_{j}h_{j}\right)=\sum_{i,j}(m_{i}n_{j})(g_{i}h_{j}),\quad
m_{i},n_{j}\in{\bf Z},g_{i},h_{j}\in\pi.$

Let $A$ be an abelian group and

$\rho\colon\thinspace\pi\to\operatorname{Aut}_{{\bf Z}}(A)$

be a homomorphism.

######  Exercise 2.8.2.

Show that representations of $\pi$ can be identified with left modules over
${\bf Z}\pi$.

Let $X$ be a path connected and locally path-connected666$X$ is locally path-
connected if for every point $x\in X$ and every open set $V$ containing $x$
there exists a path-connected open set $U$ with $x\in U\subset V$. space $X$
with a base point $x_{0}$ which admits a universal cover777A covering space of
$M$ is a space $C$ together with a continuous surjective map
$p\colon\thinspace C\to M$, such that for every $x\in M$, there exists an open
neighborhood $U$ of $x$, such that $p^{-1}(U)$ is a disjoint union of open
sets in $C$, each of which is mapped homeomorphically onto $U$ by $p$. A
connected covering space is a universal cover if it is simply connected.. Let
$\tilde{X}\to X$ be the universal cover of $X$. $\pi_{1}X$ naturally acts on
$\tilde{X}$ on the right. Then the singular chain complex888Singular
$q$–chains are linear combinations of $q$–simplices
$\textstyle\sigma\colon\thinspace\Delta^{q}=\\{(t_{0},\ldots,t_{1})\in{\bf
R}^{g+1}\mid\sum t_{i}=1,t_{i}\geq 0\text{ for all }i\\}\to X$ and form a
chain complex with the boundary map given by
$\delta(\sigma)\coloneqq\sum_{m=0}^{q}(-1)^{m}\sigma\circ f^{q}_{m},$ where
$f^{q}_{m}(t_{0},\ldots,t_{q-1})\coloneqq(t_{0},\ldots,t_{m-1},0,t_{m},\ldots,t_{q-1})$
is the face-map.  $S_{*}(\tilde{X})$ is a right ${\bf Z}\pi$-module.

######  Definition 2.8.3.

Consider a left ${\bf Z}\pi$–module $A$ for $\pi=\pi_{1}X$ and form the chain
complex

$S_{*}(X;A)=S_{*}(\tilde{X})\otimes_{{\bf Z}\pi}A.$

The homology $H_{*}(X;A)$ of this complex is called the homology of $X$ with
local coefficients in $A$. If $A$ is defined via a representation
$\rho\colon\thinspace\pi\to A$, then $H_{*}(X;A)$ is called the homology of
$X$ twisted by $\rho$ and denoted by $H_{*}(X;A_{\rho})$.

We can turn $S_{*}(\tilde{X})$ into a left ${\bf Z}\pi$–module by the standard
procedure

$g\cdot z\coloneqq z\cdot g^{-1}.$

######  Definition 2.8.4.

Consider a left ${\bf Z}\pi$–module $A$ for $\pi=\pi_{1}X$ and form the
cochain complex

$S^{*}(X;A)\coloneqq\operatorname{Hom}_{{\bf Z}\pi}(S_{*}(\tilde{X}),A).$

The cohomology $H^{*}(X;A)$ of this complex is called the cohomology of $X$
with local coefficients in $A$. If $A$ is defined via a representation
$\rho\colon\thinspace\pi\to A$, then $H^{*}(X;A)$ is called the cohomology of
$X$ twisted by $\rho$ and denoted by $H^{*}(X;A_{\rho})$.

See [18, Chapter 5] for details.

######  Exercise 2.8.5.

Convince yourself, that the usual integer cohomology of $X$ is isomorphic to
$H^{*}(X;{\bf Z}_{\rho})$ for the trivial representation
$\rho\colon\thinspace\pi_{1}X\to{\bf Z}$.

If $M$ is a closed manifold of dimension $n$ and $A$ is vector space equipped
with a non-degenerate inner product $\langle\cdot,\cdot\rangle$, then Poincaré
duality can be generalized to twisted cohomology without an extra effort

$H^{i}(M;A_{\rho})\cong H_{n-i}(M;A_{\rho}).$

Just like for the singular cohomology we have a relative version for twisted
cohomology. If $X$ is a manifold with boundary of dimension $n$, we can
generalize Poincaré-Lefschetz duality

$H^{i}(X,\partial X;A_{\rho})\cong H_{n-i}(X;A_{\rho})\quad\text{and}\quad
H^{i}(X;A_{\rho})\cong H_{n-i}(X,\partial X;A_{\rho}).$

The universal coefficient theorem can also be generalized to twisted
cohomology

$H^{i}(X;A_{\rho})\cong\operatorname{Hom}(H_{i}(X;A_{\rho}),{\bf R}).$

And certainly, we have long exact sequences in twisted homology and
cohomology.

######  Proposition 2.8.6.

Let $X$ be a manifold of dimension $2n+1$ with boundary. Then

$\dim\operatorname{im}\left[j^{*}\colon\thinspace H^{n}(X;A_{\rho})\to
H^{n}(\partial X;A_{\rho})\right]=\tfrac{1}{2}\dim(H^{n}(\partial
X;A_{\rho})).$

###### Proof.

We will suppress the coefficients in our notation. Consider the following
diagram made up out of the middle part of the long exact sequences in homology
and cohomology of the pair $(X,\partial X)$ and the Poincaré-duality
isomorphisms:

$\begin{diagram}$

Then $\dim(\operatorname{im}j^{*})=\dim(\operatorname{im}i_{*})=\dim(\ker
j_{*})$. Now recall the fact, that if $L:V\to W$ is a homomorphism between
finite-dimensional ${\bf R}$-vector spaces and $L^{*}:W^{*}\to V^{*}$ its
induced dual homomorphism, then $\ker
L\cong\operatorname{coker}L^{*}=V^{*}/\operatorname{im}L^{*}$. Then together
with the universal coefficient theorem and Poincaré duality we get

$\dim(\operatorname{im}j^{*})=\dim(\ker
j_{*})=\dim(\operatorname{coker}j^{*})=\dim(H^{n}(\partial
X))-\dim(\operatorname{im}j^{*})$

which proves $\dim(\operatorname{im}j^{*})=\frac{1}{2}\dim(H^{n}(\partial X)$.
∎

Let $\pi$ be an arbitrary group. The group cohomology $H^{*}(\pi,A_{\rho})$ of
$\pi$ twisted by $\rho$ is the cohomology of the complex given by
$C^{0}(\pi,A_{\rho})=A$, $C^{n}(\pi,A_{\rho})=\operatorname{Maps}(\pi^{n},A)$
and the boundary maps

$\displaystyle\delta^{0}(v)(g)=$ $\displaystyle\rho(g)(v)-v$
$\displaystyle\delta^{n}(f)(g_{1},\ldots,g_{n+1})=$
$\displaystyle\rho(g_{1})(f(g_{2},\ldots,g_{n+1}))$
$\displaystyle{}+\sum_{i=1}^{n}(-1)^{i}f(g_{1},\ldots,g_{i}g_{i+1},\ldots,g_{n+1})+(-1)^{n+1}f(g_{1},\ldots,g_{n}).$

The $k$–th cohomology is given by $Z^{k}(\pi,A_{\rho})/B^{k}(\pi,A_{\rho})$,
where $Z^{k}(\pi,A_{\rho})=\ker(\delta^{k})$ are the $k$–cocycles and
$B^{k}(\pi,A_{\rho})=\operatorname{im}(\delta^{k-1})$ are the
$k$–coboundaries. For example a $1$-cocycle $\xi$ is a map $\pi\to A$, which
satisfies the cocycle condition

$\xi(gh)=\xi(g)+\rho(g)\xi(h).$

Eilenberg-Maclane spaces are spaces $K(G,n)$ defined up to homotopy with the
property that all homotopy groups except for $\pi_{n}(K(G,n))=G$ are trivial.
It is shown in [32], that any compact $3$ manifold with torsion-free
fundamental group is a $K(\pi_{1}(X),1)$. It turns out that
$H^{i}(K(\pi,1),A_{\rho})=H^{i}(\pi,A_{\rho})$.

###  2.9. Algebraic sets and the Zariski tangent space

We will review necessary definitions and results from real algebraic
geometric. See [11] for more information.

For a subset $S$ of the polynomial ring ${\bf R}[x]$, $x\in{\bf R}^{n}$ we
call

${\mathcal{Z}}(S)\coloneqq\\{x=(x_{1},\ldots,x_{n})\in{\bf R}^{n}\mid
g(x)=0\text{ for all }g\in S\\}$

an algebraic set. By the Hilbert basis theorem, every ideal in ${\bf R}[x]$ is
finitely generated. In particularly the ideal generated by any subset
$S\subset{\bf R}[x]$ is generated by finite set $S^{\prime}$ so that
${\mathcal{Z}}(S)={\mathcal{Z}}(S^{\prime})$. In fact
${\mathcal{Z}}(\\{f_{1},\ldots,f_{s}\\})={\mathcal{Z}}(f)$ for
$f=\sum_{i}f^{2}_{i}$. A semi-algebraic set of ${\bf R}^{n}$ is a finite
number of unions, intersections and complements of sets $S(f_{i})$, where

$S(f)=\\{x\in{\bf R}^{n}\mid f(x)>0\\}.$

######  Exercise 2.9.1.

The collection of algebraic sets in ${\bf R}^{n}$ is closed under finite union
and arbitrary intersection. Also $\emptyset$ and ${\bf R}^{n}$ are algebraic
sets.

We can endow an algebraic set $V$ with a topology, the Zariski topology, where
the closed sets are given by the algebraic sets. Given an algebraic set $V$,
we can form the vanishing ideal

${\mathcal{I}}(V)\coloneqq\\{f\in{\bf R}[x]\mid f(x)=0\text{ for all }x\in
V\\}.$

Then one can show that $V={\mathcal{Z}}({\mathcal{I}}(V))$. The dimension of a
semi-algebraic set $V$ is defined to be the dimension of the ring of
polynomial functions on $V$

${\mathcal{P}}(V)={\bf R}[x]/{\mathcal{I}}(V),$

i.e. the maximal length of chains of prime ideals of ${\mathcal{P}}(V)$. The
coordinate ring ${\bf R}[V]$ is the ring of polynomials restricted to $V$.
$V\neq\emptyset$ is irreducible, if $V=V_{1}\cup V_{2}$ implies $V=V_{1}$ or
$V=V_{2}$. We will only consider irreducible algebraic sets, also known as
affine algebraic varieties.

For an algebraic set $V$ and $y\in V$ the tangent space can be defined as the
kernel $T_{y}V\coloneqq\ker J_{y}$ of the Jacobian matrix

$J_{y}\coloneqq\left(\frac{\partial f_{i}}{\partial
x_{j}}(y)\right)_{i,j}\colon\thinspace{\bf R}^{n}\to{\bf R}^{s},$

where ${\mathcal{I}}(V)$ is generated by $\\{f_{1},\ldots,f_{s}\\}$. For our
purpose there is a more convenient definition (due to Patrick [50]).

######  Definition 2.9.2.

Let $M$ be a smooth manifold and $V$ be a subset with the induced topology.
The tangent cone $C_{x}V$ to $V$ at $x\in V$ consists of all tangent vectors
$\dot{\gamma}(0)\in T_{x}M$ of smooth curves
$\gamma\colon\thinspace(-\epsilon,\epsilon)\to M$ with $\gamma(0)=x$ and
$\gamma([0,\epsilon))\subset V$. The tangent space $T_{x}V$ at $x\in V$ is the
linear span of $C_{q}V$.

This definition has the obvious advantage of applying to more general sets, in
particular to semi-algebraic sets.

######  Exercise 2.9.3.

Check that both definitions for the tangent space are equivalent in the case
of affine algebraic varieties.

If $\pi$ is a finitely presented discrete group and $G$ is a Lie group, then
$\operatorname{Hom}(\pi,G)$ can be considered as a subspace of the manifold
$G^{n}$, where $n$ is the number of generators. $G$ acts on
$\operatorname{Hom}(\pi,G)$ on the right via $(\rho\cdot
g)(\gamma)=g^{-1}\rho(\gamma)g$. Therefore, $\operatorname{Hom}(\pi,G)/G$ is a
subset of the $G^{n}/G$, where $G$ acts simultaneously on the right by
conjugation. If $G$ is algebraic, then $\operatorname{Hom}(\pi,G)$ is an
algebraic set. In general, $\operatorname{Hom}(\pi,G)/G$ may not even be
Hausdorff and one has to pass to the categorical quotient
$\operatorname{Hom}(\pi,G)//G$. See Sikora [58] for a detailed account for
complex reductive algebraic groups $G$.

We will compute the tangent spaces to $\operatorname{Hom}(\pi,G)$ and
$\operatorname{Hom}(\pi,G)/G$ using Definition 2.9.2 for any Lie group $G$, as
long as $\operatorname{Hom}(\pi,G)/G$ is a subspace of some manifold. It was
Weil who showed how to identify $\operatorname{Hom}(\pi,G)/G$ with
$H^{1}(\pi,\mathfrak{g}_{\rho})$ [63].

######  Theorem 2.9.4.

If $\pi$ is a finitely presented discrete group and $G$ a Lie group, such that
$\operatorname{Hom}(\pi,G)/G$ is a subspace of some manifold, then

$T_{\rho}(\operatorname{Hom}(\pi,G))=Z^{1}(\pi,\mathfrak{g}_{\rho})\quad\text{and}\quad
T_{[\rho]}(\operatorname{Hom}(\pi,G)/G)=H^{1}(\pi,\mathfrak{g}_{\rho}).$

###### Proof.

If $\pi$ is generated by $x_{1},\ldots,x_{n}$, then

$\displaystyle\operatorname{Hom}(\pi,G)$ $\displaystyle\hookrightarrow
G\times\ldots\times G$ $\displaystyle\rho$
$\displaystyle\mapsto(\rho(x_{1}),\ldots,\rho(x_{n})).$

Therefore a tangent vector at a representation
$\rho\in\operatorname{Hom}(\pi,G)$ can be considered as an element of
$T_{\rho(x_{1})}G\times\ldots\times T_{\rho(x_{n})}G$. If we want to think of
a tangent vector independently of the set of generators, we can simply
consider it as a map $\eta\colon\thinspace\pi\to TG$ with $\eta(x)\in T_{x}G$.
Now let $\rho\colon\thinspace(-\epsilon,\epsilon)\to{\bf R}^{n}$ be a path
satisfying $\rho(0)=\rho$ and
$\rho([0,\epsilon))\subset\operatorname{Hom}(\pi,G)$. By right translation we
can write $\dot{\rho}_{t}(x)=\xi_{t}(x)\rho_{t}(x)$ for
$\xi(x)\in\mathfrak{g}$. Therefore the vector $\dot{\rho}_{0}$ at $\rho$ can
be identified with a map $\xi_{0}=\xi\colon\thinspace\pi\to\mathfrak{g}$
satisfying

$\xi(xy)\rho(xy)=\left.\tfrac{d}{dt}\rho_{t}(xy)\right|_{t=0}=\left.\tfrac{d}{dt}\rho_{t}(x)\cdot\rho_{t}(y)\right|_{t=0}=\xi(x)\rho(x)\rho(y)+\rho(x)\xi(y)\rho(y),$

or equivalently the cocycle condition

$\xi(xy)=\xi(x)+\operatorname{Ad}_{\rho(x)}\xi(y).$

This shows that $T_{\rho}\operatorname{Hom}(\pi,G)\cong
Z^{1}(\pi,\mathfrak{g}_{\rho})$.

Let us turn to $T_{[\rho]}(\operatorname{Hom}(\pi,G)/G)$. A similar argument
as above shows that two tangent vectors $\xi_{1}$ and $\xi_{2}$ to
$\operatorname{Hom}(\pi,G)/G$ at $[\rho]$ associated to $\rho_{t}$ and
$g_{t}\rho_{t}g_{t}^{-1}$ respectively—which after conjugation we may assume
to be represented by tangent vectors to $\operatorname{Hom}(\pi,G)$ at
$\rho$—are equivalent if there is some $\dot{g}_{0}=v\in\mathfrak{g}$ so that

$\xi_{1}(x)\rho(x)=\left.\tfrac{d}{dt}\rho_{t}(x)\right|_{t=0}=\left.\tfrac{d}{dt}g_{t}\rho_{t}(x)g_{t}^{-1}\right|_{t=0}=v\rho(x)+\xi_{2}(x)\rho(x)-\rho(x)v.$

Therefore $\xi_{1},\xi_{2}\in Z^{1}(\pi,\mathfrak{g}_{\rho})$ are equivalent
if

$\xi_{1}(x)-\xi_{2}(x)=v-\rho(x)v\rho(x)^{-1}=\delta^{0}(v)(x),$

which shows

$T_{[\rho]}(\operatorname{Hom}(\pi,G)/G)\cong
Z^{1}(\pi,\mathfrak{g}_{\rho})/B^{1}(\pi,\mathfrak{g}_{\rho})=H^{1}(\pi,\mathfrak{g}_{\rho}).\qed$

##  Chapter 3 Connections on principal bundles

###  3.1. A few equivalent notions

If $\pi\colon\thinspace P\to M$ is a principal $G$–bundle, then for every
$p\in P$ we have a map $i_{p}\colon\thinspace G\to P$ given by the formula
$i_{p}(g)=p\cdot g$. For $X\in\mathfrak{g}$ we let $X^{*}$ be the fundamental
vector field corresponding to $X$ given by $X^{*}_{p}=di_{p}(X_{e})$.

######  Exercise 3.1.1.

$X^{*}$ is smooth and vertical, i.e. $d\pi(X^{*})=0$, and
$X^{*}_{pg}=di_{p}(X_{g})$ for $p\in P$.

######  Definition 3.1.2.

A connection on a principal $G$-bundle $P\to M$ over a manifold $M$ is a
$\mathfrak{g}$–valued 1–form $A$ on $P$ satisfying

  1. (1)

$A(X^{*})=X$ for every $X\in\mathfrak{g}$;

  2. (2)

$A$ is $G$–equivariant, i.e. $r^{*}_{g}(A)=\operatorname{Ad}_{g^{-1}}A$ for
each $g\in G$,

where $r_{g}\colon\thinspace P\to P$ denotes the right action by $g\in G$. We
denote the set of connections on $P$ by ${\mathcal{A}}_{P}$.

$G$–equivariance says that if $Y$ is a vector field on $P$ and $g\in G$, then
$A(dr_{g}(Y))=\operatorname{Ad}_{g^{-1}}(A(Y))$. Locally, a 1–form on an
$n$–dimensional manifold $M$ with values in $\mathfrak{g}$ has the expression

$(dx_{1}\otimes A_{1})+\ldots+(dx_{n}\otimes A_{n}),$

where the $x_{i}$ are coordinates on $M$ and the $A_{i}$ are elements of
$\mathfrak{g}$. Given a vector bundle $E\to M$, $\Omega^{p}(E)$ will denote
the $p$–forms with values in $E$, i.e. smooth sections of the vector bundle

$\bigwedge^{p}T^{*}M\otimes E\to M,$

where the symbols denote the wedge and tensor products of the bundles. A
connection is therefore an element of $\Omega^{1}(P;\mathfrak{g})$, where

$\Omega^{k}(P;\mathfrak{g}):=\Omega^{k}(P\times\mathfrak{g})$

are the $\mathfrak{g}$–valued $k$–forms on $P$. Denote the subspace of
$G$–equivariant $\mathfrak{g}$–valued $k$–forms on $P$ by
$\Omega^{k}(P;\mathfrak{g})^{G}$.

Alternatively consider a $G$–invariant horizontal distribution on $P\to
M^{n}$, i.e. an $n$–dimensional distribution $H$ on $TP$ satisfying:

  1. (1)

$H_{p}\subset T_{p}P\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}T_{\pi(p)}M$ is an isomorphism for each $p\in P$;

  2. (2)

The distribution is $G$–invariant111The terminology $G$–equivariant would also
be appropriate, i.e. $H_{pg}=d(r_{g})H_{p}$;

  3. (3)

$H_{p}$ varies smoothly with $p$.

The $H_{p}$ are called the horizontal subspaces.

######  Exercise 3.1.3.

The map from the connections on $P$ to the $G$–invariant horizontal
distributions on $P$ given by $A\mapsto\ker A$ is a bijection. Use the fact
that the choice of $H_{p}$ yields the decomposition $T_{p}P=H_{p}\oplus
V_{p}$, where $V_{p}$ is the vertical subspace spanned by all the fundamental
vector fields.

Therefore, there is a slick way to think about connections: A connection is a
choice of a $G$–equivariant splitting of the family

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{di_{p}}$$\scriptstyle{A}$$\textstyle{T_{p}P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d\pi_{p}}$$\scriptstyle{(d\pi|_{H_{p}})^{-1}}$$\textstyle{T_{\pi(p)}M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

of short exact sequences parametrized by $p\in P$. This can be viewed as a
splitting of the Atiyah sequence [7]. As we have seen above, this splitting
can be given by a left-inverse of $i_{p}$ (retraction), namely $A$, or by a
right-inverse of $d\pi_{p}$ (section), namely $(d\pi|_{H_{p}})^{-1}$.

From Exercise 3.1.3 we see that a connection on $P$ determines a unique way to
lift tangent vectors from $M$: If $X_{x}\in T_{x}M$ let $\tilde{X}_{p}\in
T_{p}P$ be the unique vector in $H_{p}$ lying over $X_{x}$, where
$p\in\pi^{-1}(x)$. The lift $\tilde{X}$ is called the horizontal lift of $X$.
Since we can lift vectors from $M$ to $P$, it seems believable that we can
lift a path in such a way that the tangent vectors to the lifted path are the
horizontal lifts of the tangent vectors to the path in $M$. This is another
characterization of a connection and will be discussed in Section 3.3.

Our original definition can be modified a little by using the Maurer Cartan
form. This will be discussed in Section 3.7

If we have a representation we can consider the vector bundle associated to a
principal bundle and consider its covariant derivative, which is yet another
notion of the connection and will be discussed in Section 3.4.

###  3.2. Structure of the space of connections

If $P$ is trivializable, then a trivialization $P\cong M\times G$ gives a
decomposition $T_{(x,g)}P=T_{x}M\oplus T_{g}G$. Thus a trivialization gives us
a choice of horizontal and vertical subspaces, this choice is called the
trivial connection (with respect to this trivialization). The corresponding
$\mathfrak{g}$–valued 1–form on $P$ takes a vector $X=(X_{1},X_{2})\in
T_{x}M\oplus T_{g}G$ to $dl_{g^{-1}}X_{2}\in T_{e}G=\mathfrak{g}$ (or
equivalently to the left-invariant vector field $h\mapsto dl_{hg^{-1}}X_{2}$).
A different trivialization of the bundle determines a different trivial
connection.

In the case of a trivializable bundle we can simplify the notion of a
connection.

######  Proposition 3.2.1.

Let $P$ be a trivializable principal $G$-bundle over a manifold $M$. A
trivialization of $P$ identifies the space of connections ${\mathcal{A}}_{P}$
on $P$ with $\mathfrak{g}$–valued $1$-forms $\Omega^{1}(M;\mathfrak{g})$ on
$M$.

###### Proof.

Let $A$ be a connection on $P$. By trivializing $P=M\times G$, we can
decompose the tangent space of $P$ at $p=(x,g)\in P$ as $T_{p}P=T_{x}M\times
T_{g}G$.

Let $s:M\to P=M\times G$ be the section $s(x):=(x,e)$. Then we can define an
affine homomorphism

$\displaystyle\Psi:{\mathcal{A}}_{P}$
$\displaystyle\to\Omega^{1}(M)\otimes\mathfrak{g}$ $\displaystyle A$
$\displaystyle\to s^{*}(A).$

On the other hand given $\tilde{A}\in\Omega^{1}(M;\mathfrak{g})$, where we
think of $\mathfrak{g}$ as $T_{e}G$, define a $\mathfrak{g}$–valued $1$-form
$\Phi(\tilde{A}):=A$ on $P=M\times G$ where

$A_{(x,g)}(v_{x},w_{g})=\tilde{A}_{x}(v_{x})+d(l_{g^{-1}})w_{g}.$

Then for $X\in\mathfrak{g}$ and $p=(x,g)\in P$ we have
$X_{p}^{*}=d(i_{p})X=(0_{x},d(l_{g})X)$ and thus

$A_{p}(X^{*})=\tilde{A}_{x}(0_{x})+X=X.$

Since $r_{h}(x,g)=(x,gh)$ we have $d(r_{h})(v_{x},w_{g})=(0,d(r_{h})w_{g})$
and therefore

$\displaystyle r_{h}^{*}A(v_{x},w_{g})$
$\displaystyle=A(0_{x},d(r_{h})w_{g})=\tilde{A}(0_{x})+d(l_{(gh)^{-1}})d(r_{h})w_{g}$
$\displaystyle=\operatorname{Ad}_{h^{-1}}\tilde{A}(0_{x})+d(l_{h^{-1}})d(r_{h})d(l_{g^{-1}})w_{g}=\operatorname{Ad}_{h^{-1}}A(v_{x},w_{g}).$

This way we have a homomorphism
$\Psi:\Omega^{1}(M;\mathfrak{g})\to{\mathcal{A}}_{P}$.

Furthermore, the maps $\Phi$ and $\Psi$ are inverses of each other, because

$(\Psi\circ\Phi(\tilde{A}))v_{x}=s^{*}(A_{(x,g)})(v_{x})=A_{(x,g)}(v_{x},0_{g})=\tilde{A}_{x}v_{x}$

and

$(\Phi\circ\Psi(A))(v_{x},w_{g})=\Psi
A(v_{x})+d(l_{g^{-1}})w_{g}=A(v_{x},0_{g})+A(0_{x},w_{g})=A(v_{x},w_{g}).$

That is, ${\mathcal{A}}_{P}\cong\Omega^{1}(M;\mathfrak{g})$ as (affine) vector
spaces. ∎

We have already mentioned that a different trivialization changes the
connection. The following exercise describes this change.

######  Exercise 3.2.2.

Choose a connection in a trivializable bundle $P\to M$. Trivialize $P$ by
choosing a section $s\colon\thinspace M\to P$ so that the connection pulls
back to a 1–form $A\in\Omega^{1}(M;\mathfrak{g})$. Choose a map
$h\colon\thinspace M\to G$ and use $h$ to define a new secion by the formula
$s_{h}(x)=s(x)\cdot h(m)$. Then in this trivialization the connection has the
form

$\operatorname{Ad}_{h^{-1}}A+d(l_{h^{-1}})dh.$

This means that if $X\in T_{x}M$, then evaluating the pullback of the
connection using $s_{h}$ on $X$ yields

$\operatorname{Ad}_{h^{-1}(x)}A(X)+d(l_{h^{-1}(x)})dh(X).$

Notice that $dh(X)\in T_{h(x)}G$. Prove and use the product rule

$\left.\frac{d}{dt}\right|_{t=0}s(\alpha_{t})h(\alpha_{t})=d(r_{h(\alpha_{0})})ds(\dot{\alpha}_{0})+d(i_{s(x)})dh(\dot{\alpha}_{0})$

for a path $\alpha$ with derivative $X$, where $i_{p}(g)=p\cdot g$ and
$r_{g}(p)=p\cdot g$.

We have not yet discussed the existence of connections on general principal
bundles, but it easily follows from the existence of connections on trivial
bundles by the fact that $tA_{0}+(1-t)A_{1}$ is a connection if $A_{0}$ and
$A_{1}$ are and by using a partition of unity. The collection of all
connections on $P$ forms a topological space ${\mathcal{A}}$ which is a subset
of the vector space of all $\mathfrak{g}$–valued 1–forms on $P$. Even though
${\mathcal{A}}_{P}$ is not a linear subspace, since the sum of two connections
violates the first condition, it is, however, an affine space.

######  Proposition 3.2.3.

A choice of base connection determines an isomorphism of ${\mathcal{A}}_{P}$
with $\Omega^{1}(\operatorname{Ad}_{P})$.

###### Proof.

Fix a connection $A_{0}$. By Exercise 2.7.8 sections of $P\times_{\rho}V$ are
the same as equivariant maps from $P$ to $V$. Let $A$ be another connection.
Given a vector field $X$ on $M$, we can construct a function
$\bar{A}_{X}\colon\thinspace P\to\mathfrak{g}$ by the formula:

$\bar{A}_{X}(p)=A(\tilde{X}_{p}),$

where $\tilde{X}_{p}$ is the horizontal lift of $X$ to $T_{p}P$ with respect
to $A_{0}$. The properties of a connection show that $\bar{A}_{X}$ is indeed
an equivariant map from $P$ to $\mathfrak{g}$, and therefore $\bar{A}_{X}$
determines an element of $\Omega^{1}(\operatorname{Ad}_{P})$.

Conversely, let $\bar{A}\in\Omega^{1}(\operatorname{Ad}_{P})$. Define

$A(Y)=\bar{A}(Y_{1})(p)+Y_{2}\quad\text{for }Y_{2}\in T_{p}P,$

where $Y=Y_{1}+Y_{2}$ is the decomposition into horizontal and vertical part
with respect to $A_{0}$. Thus the choice of $A_{0}$ identifies
${\mathcal{A}}_{P}$ with $\Omega^{1}(\operatorname{Ad}_{P})$. ∎

######  Exercise 3.2.4.

We have seen that a trivialized bundle has a natural trivial connection. We
can use it as a base connection to identify ${\mathcal{A}}_{P}$ with
$\Omega^{1}(\operatorname{ad}g)$, which in turn is naturally isomorphic
$\Omega^{1}\otimes\mathfrak{g}$ for a trivialized bundle $P$. Show that this
identification is the same as the one used in Proposition 3.2.1.

###  3.3. Lifting paths

Let $A$ be a connection in a principal bundle $P\to M$. Let
$\alpha\colon\thinspace I\to M$ be a smooth path and $p_{0}\in P$ a point in
the fiber over $\alpha_{0}$.

######  Proposition 3.3.1.

There exists a unique lifting $\beta\colon\thinspace I\to P$ so that for each
$t\in I$, the $\dot{\beta}_{t}$ is the horizontal lift of $\dot{\alpha}_{t}$
with starting point $p_{0}$. The lift $\beta$ is called the horizontal lift of
$\alpha$.

###### Proof.

It is easy to see, that there exists some lift $\gamma$ of $\alpha$ to $P$
(e.g. by the lifting property or by trivializing the principal bundle over
contractible components along the path). Then any path $\beta_{t}$ is of the
form $\gamma_{t}\cdot g_{t}\in P$ for some path $g\colon\thinspace I\to G$.
Now $\beta$ is a horizontal lift if and only if $A(\dot{\beta}_{t})=0$ for all
$t\in I$. We can apply the product rule to get

$A(\dot{\beta}_{t})=A((r_{g_{t}})_{*}\dot{\gamma}_{t}+(i_{\gamma_{t}})_{*}\dot{g}_{t})=\operatorname{Ad}_{g_{t}^{-1}}A(\dot{\gamma}_{t})+(l_{g_{t}^{-1}})_{*}\dot{g}_{t}.$

Therefore $\beta$ is a horizontal lift if and only if
$A(\dot{\gamma}_{t})=-(r_{g_{t}^{-1}})_{*}\dot{g}_{t}$. To find the
appropriate $g$ with starting point $g_{0}$ is just a matter of solving an
ordinary differential equation in $G$. ∎

If $\alpha$ is a loop, i.e. $\alpha(0)=\alpha(1)$, then the starting point
$p_{0}$ and the endpoint of the lift $\beta$ live in the same fiber and we can
find an element $g\in G$ with $\beta(1)=\beta(0)\cdot g$. We call
$\operatorname{hol}_{A}(\alpha,p_{0}):=g$ the holonomy of $A$ along $\alpha$
with respect to the base point $p_{0}$. Therefore we get the following from
Proposition 3.3

######  Corollary 3.3.2.

Given a base point $p_{0}\in\pi^{-1}(x_{0})$, any connection $A$ defines a map
$\operatorname{hol}_{A}\colon\thinspace\operatorname{Loops}(M,x_{0})\to G$
called the holonomy map.

This holonomy map does not define a map on $\pi_{1}$, unless the connection is
flat, a term we will define later. One can recapture the connection by knowing
what the horizontal lifts of all paths are, just define the horizontal
subspace at $p\in P$ to be the space spanned by the derivatives of the
horizontal lifts of all paths throough $\pi(p)$. One can therefore give yet
another definition of a connection in a principal bundle to be a coherent way
to lift paths in $M$ so that equivariance is satisfied.

###  3.4. Connections in vector bundles

We now turn to connections in vector bundles. One should think of these as a
way to differentiate sections of vector bundles in the direction of a vector
field on the base manifold. For example, given a function
$f\colon\thinspace{\bf R}^{n}\to{\bf R}$, and a vector field $X$ on ${\bf
R}^{n}$, then we can take the derivative of $f$ along the vector field to get
another function $df(X)\colon\thinspace{\bf R}^{n}\to{\bf R}$. Similarly, if
we have a (vector valued) function $f\colon\thinspace{\bf R}^{n}\to{\bf
R}^{p}$, and a vector field $X$ on ${\bf R}^{n}$ we can apply the preceding
procedure componentwise to obtain a new function $df(X)\colon\thinspace{\bf
R}^{n}\to{\bf R}^{p}$. Again this can be done by replacing ${\bf R}^{n}$ by
any manifold $M$, thus there is a natural way to take derivatives of functions
$M\to{\bf R}^{p}$ along vector fields. But functions are just sections of
trivial bundles, and the general notion of a connection in a vector bundle is
a way to do this when the bundle is not trivial.

######  Definition 3.4.1.

An affine connection, also called covariant derivative on $E$, is a linear
mapping

$\nabla\colon\thinspace\Omega^{0}(E)\to\Omega^{1}(E)$

satisfying the Leibniz rule

$\nabla(f\varphi)=df\otimes\varphi+f\nabla\varphi\quad\text{for }f\in
C^{\infty}(M,{\bf R})\text{ and }\varphi\in\Omega^{0}(E).$

Here we use that $\Omega^{0}(E)$ is isomorphic to $\Gamma(E)$ via the map
$f\otimes\varphi\mapsto f\varphi$, since $\Lambda^{0}T^{*}M$ is just the
trivial line bundle.

If $X$ is a vector field on $M$, then we will write $\nabla_{X}(\varphi)$ for
the evaluation $(\nabla\varphi)(X)$. If $E$ is associated to a principal
$U(n)$ or $\text{\it{SO}}(n)$ bundle via the standard representation, then $E$
has a fiberwise inner product $(\cdot,\cdot)$. We say $\nabla$ is compatible
with the metric if

$d(\varphi,\theta)=(\nabla\varphi,\theta)+(\varphi,\nabla\theta).$

This expression makes sense since $(\varphi,\theta)$ is a smooth function on
$M$, thus $d(\varphi,\theta)\in\Omega^{1}(M)$. Also
$(\nabla\varphi,\theta)\in\Omega^{1}(M)$ since we can evaluate it on a vector
field $X\colon\thinspace(\nabla\varphi,\theta)(X)=(\nabla_{X}\varphi,\theta)$.
Compatibility with the metric refers to the metric on the vector bundle, not
the Riemannian metric on $M$.

######  Example 3.4.2.

Let $E=M\times{\bf C}^{n}$, the trivial ${\bf C}^{n}$ bundle. Sections of $E$
are simply functions $\varphi\colon\thinspace M\to{\bf C}^{n}$. Let
$d\colon\thinspace\Omega^{0}\to\Omega^{1}$ denote the exterior derivative. It
is defined for complex valued functions on $M$. Then $d$ defines the trivial
connection on $E$:

$\displaystyle d\colon\thinspace\Omega^{0}(E)$ $\displaystyle\to\Omega^{1}(E)$
$\displaystyle\varphi=(\varphi_{1},\ldots,\varphi_{n})$
$\displaystyle\mapsto(d\varphi_{1},\ldots,d\varphi_{n}).$

Using this example and a partition of unity we see that any vector bundle
admits an affine connection.

For $E=M\times{\bf C}^{n}$ and $A\in\Omega^{1}(M)\otimes\mathfrak{g}$, where
$G\subset GL_{n}({\bf C})$, we can consider

$\displaystyle d^{A}\colon\thinspace$
$\displaystyle\Omega^{0}(E)\to\Omega^{1}(E)$
$\displaystyle\varphi=(\varphi_{1},\ldots,\varphi_{m})$
$\displaystyle\mapsto(d\varphi_{1},\ldots,d\varphi_{m})+A\varphi.$

The term $A\varphi$ is a 1–form with values in $E$, since given a vector field
$X$, $A(X)\in\mathfrak{g}$ which acts on sections of $E$ by left matrix
multiplication.

######  Exercise 3.4.3.

Show that $d^{A}$ is an affine connection on $E$. If $G=U(n)$, show that
$d^{A}$ is compatible with the (trivial) Hermitian metric on $E$ if and only
if $A\in\Omega^{1}(M)\otimes\mathfrak{u}(n)$.

######  Example 3.4.4.

We will now show that a connection on any $\text{\it{GL}}(n,{\bf C})$–vector
bundle $E$ can be written locally as $d+A$ for some $\mathfrak{gl}(n,{\bf
C})$–valued 1–form $A$ on $M$. A convenient way to show this is to use local
frames. Let $\\{e_{i}\\}$ be a local frame for a vector bundle $E$. Let
$\nabla$ be a covariant derivative and let $A_{r,s}$ be the matrix of 1–forms
determined by the expression

$\nabla e_{s}=\sum_{r}A_{r,s}\otimes e_{r}.$

(If the structure group of $E$ is $O(n)$ or $U(n)$ and the $e_{i}$ are
orthogonal then $A_{r,s}$ is a $\mathfrak{o}(n)$– or a
$\mathfrak{u}(n)$–valued form if $\nabla$ is compatible with the metric.) Now
if $\varphi\in\Gamma(E)$, we can write $\varphi=\sum\varphi_{i}e_{i}$ with
$\varphi_{i}\in C^{\infty}$ and so we must have

$\displaystyle\nabla\varphi$ $\displaystyle=\sum
d\varphi_{r}e_{r}=\sum_{r}(d\varphi_{r}\otimes e_{r}+\varphi_{r}\nabla e_{r})$
$\displaystyle=\sum_{r}(d\varphi_{r}\otimes
e_{r}+\varphi_{r}\sum_{s}A_{s,r}\otimes e_{s})$
$\displaystyle=\sum_{r}(d\varphi_{r}+\sum_{s}A_{r,s}\varphi_{s})\otimes
e_{r}.$

Writing $\varphi$ as a column vector, we can write this as

$\nabla\varphi=(d+A)\varphi.$

Hence in local coordinates any connection is of the form $d+A$.

This example hints (strongly) at the relationship between the definition in
principal bundles and in vector bundles. We saw that in a trivial principal
bundle, a connection is determined by a Lie algebra valued 1–form on $M$. The
same is true in the preceding example. One sees a local equivalence of the two
notions of connections. In the next section we will give a global
identification of these two notions, without any reference to a trivialization
of the bundle.

######  Example 3.4.5.

The Fundamental Theorem of Riemannian Geometry states that every Riemannian
manifold possesses a unique connection on its tangent bundle which satisfies
two conditions (compatible with the metric and torsion free222$\nabla$ is
called torsion-free if $\nabla_{X}Y-\nabla_{Y}X=[X,Y]$ for all vector fields
$X$ and $Y$ on $M$). Thus invariants of Riemannian manifolds can be
constructed by working with this connection called the Riemannian or Levi-
Civita connection. It should be stressed that the Levi-Civita connection only
depends on the metric.

###  3.5. Equivalence of connections in vector bundles and principal bundles

We have seen that the frame bundle of a vector bundle $E$ is a principal
bundle $P$, so that the vector bundle associated to $P$ via the defining
representation is $E$ again. Therefore, principal bundles are more general
than vector bundles and we do not loose any generality if we always start with
a principal bundle. We will see that a connection in a principal bundle always
gives us a connection in an associated vector bundle. Furthermore, if we
restrict ourselves to principal $G$-bundles where $G\subset GL(n,{\bf C})$ or
$G\subset GL(n,{\bf R})$, the notions of connections are equivalent.

Let $E=P\times_{\rho}V\to M$ be a vector bundle associated to a principal
$G$–bundle $P\to M$ and $\rho\colon\thinspace G\to GL(V)$ a representation of
$G$. We saw that the sections of $E$ are $G$–equivariant maps
$\varphi\colon\thinspace P\to V$. Notice that for a vector space $V$, the
tangent space $T_{v}V$ is canonically identified with $V$ itself.

Let $A$ be a connection in the principal bundle $P\to M$; we think of $A$ as a
horizontal distribution. Given a $G$–equivariant map $\varphi\colon\thinspace
P\to V$ and a vector field $X$ on $M$, we want to construct another
$G$–equivariant map $\nabla^{A}_{X}\varphi\colon\thinspace P\to V$. Given
$p\in P$ let $\tilde{X}_{p}$ be the unique horizontal lift of $X_{\pi(p)}$ to
$T_{p}P$ given by the connection. Define $(\nabla^{A}_{X}(\varphi))(p)$ to be
$d\varphi(\tilde{X}_{p})\in T_{v}V=V$. This construction yields a bijection
between connections in principal $\text{\it{GL}}(n,{\bf C})$–bundles and
connections in $\text{\it{GL}}(n,{\bf C})$–vector bundles associated via the
defining representation. Naturally, this will only be a bijection, when we can
reconstruct the principal bundle from the vector bundle, e.g. by considering
the frame bundle.

######  Exercise 3.5.1.

Verify that the above construction defines a bijection between connections in
principal $U(n)$– or $O(n)$–bundles and affine connections in ${\bf R}^{n}$–
or ${\bf C}^{n}$–vector bundles compatible with the metric.

Path lifting in a principal bundle corresponds to parallel transport of frames
in a vector bundle. If I choose a frame in the fiber of $E$ over $m\in M$ it
is easy to see that lifting a path $\alpha_{t}$ starting at $p\in P$ gives me
a way to choose a frame in the fiber over each point of the path, i.e. it
hands me a trivialization of the pullback bundle $\alpha^{*}(E)$ over an
intervall. This is one of the older definitiions of a connection, the “repère
mobile” (moving frame) of Élie Cartan.

###  3.6. Extending the covariant derivatives

We can extend an affine connection $\nabla$ in a vector bundle $E$ to

$\nabla\colon\thinspace\Omega^{k}(E)\to\Omega^{k+1}(E)$

by forcing the Leibniz rule to hold; so

$\nabla(\omega\otimes\varphi)=d\omega\otimes\varphi+(-1)^{k+1}\omega\wedge\nabla\varphi\quad\text{for
}\omega\in\Omega^{k}(M)\text{ and }\varphi\in\Gamma(E).$

The resulting sequence

$\cdots\to\Omega^{k-1}(E)\stackrel{{\scriptstyle\nabla}}{{\to}}\Omega^{k}(E)\stackrel{{\scriptstyle\nabla}}{{\to}}\Omega^{k+1}(E)\to\cdots$

is not necessarily a complex. In fact, we will see in the Section 3.8 that
this sequence is a complex if and only if $\nabla$ is flat.

There are two special cases of this which are particularly important, namely
the bundle associated to a principal bundle $P\to M$ via the defining
representation and the adjoint bundle. If $E$ is the bundle associated to $P$
via the defining representation (where $G$ is a subgroup of
$\text{\it{GL}}(n,{\bf R})$ or $\text{\it{GL}}(n,{\bf C})$) then the adjoint
bundle is a sub-vector bundle $\operatorname{Hom}(E,E)$, since
$\operatorname{Hom}(E,E)=P\times_{\operatorname{Ad}}\mathfrak{gl}(n)$. If you
choose a local basis of sections $\\{e_{i}\\}$ of $E$, then the covariant
derivative on $E$ has the form $d^{A}=d+A$ for some
$A\in\Omega^{1}\otimes\mathfrak{g}$, as explained in Section 3.4. Furthermore
if we fix a connection $\nabla$, then any other connection can be written as
$\nabla^{A}=\nabla+A$ globally for some
$A\in\Omega^{1}(\operatorname{Ad}_{P})$.

Suppose we are given two differential forms $\alpha\in\Omega^{p}(E)$ and
$\beta\in\Omega^{q}(E^{\prime})$, then a priori we have
$\alpha\wedge\beta\in\Omega^{p+q}(E\otimes E^{\prime})$ by extending
$(\alpha\otimes\varphi)\wedge(\alpha^{\prime}\otimes\varphi^{\prime}):=(\alpha\wedge\alpha^{\prime})\otimes(\varphi\otimes\varphi^{\prime})$
linearly. But if we are given a section of $\operatorname{Hom}(E\otimes
E^{\prime},E^{\prime\prime})$, i.e. a smoothly varying bilinear form
$\Phi\colon\thinspace E_{x}\times E^{\prime}_{x}\to E^{\prime\prime}_{x}$ on
the fibers over $x\in M$, then using this form we may define the product
$\Phi(\alpha\wedge\beta)\in\Omega^{p+q}(E^{\prime\prime})$ by extending
$\Phi((\alpha\otimes\varphi)\wedge(\alpha^{\prime}\otimes\varphi^{\prime})):=(\alpha\otimes\alpha^{\prime})\otimes\Phi(\varphi,\varphi^{\prime})$
linearly.

Typically $E=E^{\prime}=E^{\prime\prime}$ is the Lie algebra bundle of some
principal $G$-bundle. We will write
$[\alpha\wedge\beta]=\Phi(\alpha\wedge\beta)$, when the $\Phi$ is the Lie
bracket (see Exercise 2.7.7), and $\alpha\wedge\beta=\Phi(\alpha\wedge\beta)$,
when $\Phi$ is ordinary matrix multiplication for a matrix group $G$ or
evaluation $\operatorname{Hom}(E,E)\otimes E\to E$. Notice that for
$\alpha\in\Omega^{p}(E)$ and $\beta\in\Omega^{q}(E)$ where $E$ is a vector
bundle with fiber isomorphic to a Lie algebra $\mathfrak{g}$ (e.g.
$E=\operatorname{Ad}_{P}$) it is easy to see that the Lie-bracket satisfies

(3.6.1) $[\alpha\wedge\beta]=-(-1)^{pq}[\beta\wedge\alpha]$

as well as the graded Jacobi identity

(3.6.2)
$(-1)^{pr}[\alpha\wedge[\beta\wedge\gamma]]+(-1)^{pq}[\beta\wedge[\gamma\wedge\alpha]]+(-1)^{qr}[\gamma\wedge[\alpha\wedge\beta]]$

for $\alpha\in\Omega^{p}(E),\beta\in\Omega^{q}(E),\gamma\in\Omega^{r}(E).$

######  Example 3.6.1.

Let $A$ be a connection in a principal $G$–bundle $P\to M$ with
$G\subset\text{\it{GL}}(n,{\bf C})$ and let $E$ be the vector bundle
associated to $P$ via the defining representation. By Example 3.4.4 locally
$\nabla^{A}=d+A$, if we write $\varphi\in\Gamma(E)$ as a column vector.
Therefore the induced connection
$d^{A}\colon\thinspace\Omega^{1}(E)\to\Omega^{2}(E)$ is satisfies:

$\displaystyle d^{A}(\omega\otimes\varphi)$
$\displaystyle=d\omega\otimes\varphi-\omega\wedge\nabla^{A}\varphi$
$\displaystyle=d\omega\otimes\varphi-\omega\wedge\varphi-\omega\wedge
A\varphi$ $\displaystyle=d(\omega\otimes\varphi)+A\varphi\wedge\omega$
$\displaystyle=d(\omega\otimes\varphi)+A\wedge(\omega\otimes\varphi).$

Therefore locally $d^{A}\varphi=d\varphi+A\wedge\varphi$.

######  Exercise 3.6.2.

Show that the induced connection
$d^{A}\colon\thinspace\Omega^{1}(\operatorname{Hom}(E,E))\to\Omega^{2}(\operatorname{Hom}(E,E))$
is given by the formula

$B\mapsto dB+[A\wedge B].$

###  3.7. The Maurer-Cartan form

In this section we will see yet another equivalent, however only slightly
different definition for a connection.

######  Definition 3.7.1.

The Maurer-Cartan form $\theta$ on $G$ is the unique $\mathfrak{g}$–valued
1–form on $G$ which assigns to each vector its left-invariant extension.

Notice that for every smooth vector field $X$ we get a smooth map

$\displaystyle\theta(X)\colon\thinspace G$ $\displaystyle\to\mathfrak{g}$
$\displaystyle g$ $\displaystyle\mapsto\theta(X_{g}).$

######  Exercise 3.7.2.

Check that $\theta(X_{g})_{h}=dl_{hg{-1}}(X_{g})$ and that it satisfies

  1. (1)

$l_{g}^{*}\theta=\theta$;

  2. (2)

$r_{g}^{*}\theta=\operatorname{Ad}_{g^{-1}}\theta$.

######  Proposition 3.7.3.

The Maurer-Cartan form satisfies the Maurer-Cartan equation

$d\theta+\tfrac{1}{2}[\theta\wedge\theta]=0.$

###### Proof.

By Exercise 3.7.2 we have for a left-invariant vector fields $X$

$\theta(X_{g})_{h}=dl_{hg^{-1}}(X_{g})=X.$

Therefore the map $\theta(X)\colon\thinspace G\to\mathfrak{g}$ is constant and
$d(\theta(X))=0$. Furthermore we have for left-invariant vector fields $X$ and
$Y$

$\theta([X,Y])=[X,Y]=[\theta(X),\theta(Y)],$

which implies

$\displaystyle d\theta(X,Y)$
$\displaystyle=X(\theta(Y))-Y(\theta(X))-\theta([X,Y])$
$\displaystyle=d(\theta(Y))(X)-d(\theta(X))(Y)-[\theta(X),\theta(Y)]$
$\displaystyle=-[\theta(X),\theta(Y)].$

The evaluation of either side at $g\in G$ does not rely on the fact that $X$
and $Y$ are left-invariant vector fields.and therefore
$d\theta(X,Y)+[\theta(X),\theta(Y)]=0$ holds for arbitrary smooth vector
fields. For $\alpha_{i}\in\Omega^{1}(G)$ and $\varphi_{i}\in\mathfrak{g}$,
$i=1,2$, we have

$\displaystyle[\alpha_{1}(X)\otimes\varphi_{1},$
$\displaystyle\alpha_{2}(Y)\otimes\varphi_{2}]+[\alpha_{2}(X)\otimes\varphi_{2},\alpha_{1}(Y)\otimes\varphi_{1}]$
$\displaystyle=\alpha_{1}(X)\alpha_{2}(Y)(\varphi_{1}\varphi_{2}-\varphi_{2}\varphi_{1})+\alpha_{2}(X)\alpha_{1}(Y)(\varphi_{2}\varphi_{1}-\varphi_{1}\varphi_{2})$
$\displaystyle=\alpha_{1}\wedge\alpha_{2}(X,Y)\varphi_{1}\varphi_{2}+\alpha_{2}\wedge\alpha_{1}(X,Y)\varphi_{2}\varphi_{1}$
$\displaystyle=\alpha_{1}\wedge\alpha_{2}(X,Y)[\varphi_{1},\varphi_{2}]$
$\displaystyle=[(\alpha_{1}\otimes\varphi_{1})\wedge(\alpha_{2}\otimes\varphi_{2})](X,Y).$

This implies $[\theta(X),\theta(Y)]=\tfrac{1}{2}[\theta\wedge\theta](X,Y)$ and
completes the proof. ∎

Let $P\stackrel{{\scriptstyle\pi}}{{\to}}M$ be a principal $G$–bundle and
consider $p\in P_{x}$ for some $x\in M$, where $P_{x}=\pi^{-1}(x)$ is the
fiber over $x$. The map $i_{p}\colon\thinspace G\hookrightarrow P_{x}$ given
by $i_{p}g=p\cdot g$ induces the $\mathfrak{g}$–valued 1–form
$\theta_{x}:=i_{p}^{*}\theta$ on $P_{x}$ which satisfies

$r_{g}^{*}(\theta_{x})=\operatorname{Ad}_{g^{-1}}\theta_{x}\quad\text{and}\quad
d\theta_{x}+[\theta_{x}\wedge\theta_{x}]=0.$

######  Exercise 3.7.4.

Show that replacing the first condition in Definition 3.1.2 by

$j_{x}^{*}A=\theta_{x}\quad\text{for all }x\in M,$

where $j_{x}\colon\thinspace P_{x}\hookrightarrow P$, yields an equivalent
definition of a connection.

###  3.8. Curvature

Let $\nabla$ be a connection in a vector bundle $E$.

######  Theorem 3.8.1.

The composition

$\Omega^{0}(E)\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}\Omega^{1}(E)\stackrel{{\scriptstyle\nabla}}{{\longrightarrow}}\Omega^{2}(E)$

is a differential operator of order 0. More precisely, there exists an
$F^{\nabla}\in\Omega^{2}(\operatorname{Hom}(E,E))$ such that

$\nabla\nabla\varphi=F^{\nabla}\cdot\varphi.$

$F^{\nabla}$ is called the curvature of the affine connnection $\nabla$.

###### Proof.

We give a local proof. Choose a local frame $\\{e_{i}\\}$. In this frame the
connection has the form $\nabla=d+A$ by Example 3.4.4. Let $\varphi$ be a
section and write it as a column vector. Then by Examples 3.4.4 and 3.6.1

$\displaystyle\nabla\nabla\varphi$ $\displaystyle=\nabla(d\varphi+A\varphi)$
$\displaystyle=d(d\varphi+A\varphi)+A\wedge(d\varphi+A\varphi)$
$\displaystyle=d^{2}\varphi+d(A\varphi)+A\wedge d\varphi+A\wedge(A\varphi)$
$\displaystyle=A\wedge d\varphi+dA\wedge\varphi-A\wedge d\varphi+(A\wedge
A)\varphi$ $\displaystyle=(dA+A\wedge A)\varphi$
$\displaystyle=(dA+\tfrac{1}{2}[A\wedge A])\varphi$

where the last equality follows from (3.6.1) and

$A\wedge A=\sum_{k}A_{r,k}\wedge A_{k,s}.\qed$

######  Exercise 3.8.2.

Give a global proof of Theorem 3.8.1:

  1. (1)

Show that $\nabla\nabla(f\varphi)=f\nabla\nabla(\varphi)$ for $f\in
C^{\infty}(M)$ and $\varphi\in\Omega^{0}(M)$.

  2. (2)

Let $L\colon\thinspace\Omega^{0}(E)\to\Omega^{2}(E)$ be any
$C^{\infty}(M)$–linear map. Then there is an
$\hat{L}\in\Omega^{2}(\operatorname{Hom}(E,E))$ such that
$L(\varphi)=\hat{L}\wedge\varphi$ for all $\varphi\in\Omega^{0}(E)$.

Now we are ready to define the curvature of connections in principal
$G$–bundles.

######  Definition 3.8.3.

Let $A$ be a connection on a principal $G$–bundle $P\to M$ and consider the
induced affine connection $d^{A}$ in the adjoint bundle $\operatorname{Ad}P$
as described in Section 3.5. The curvature $F^{A}$ of a connection $A$ in a
principal $G$–bundle $P\to M$ is defined to be the local invariant
$F^{d^{A}}$. We will see below that the curvature $F^{A}$ of $d^{A}$ can be
viewed as an element of $\Omega^{2}(\operatorname{Ad}_{P})$.

We denote by $\tilde{d}^{A}$ the pull-back of $d^{A}$ to the trivial vector
bundle $P\times\mathfrak{g}=\pi^{*}(\operatorname{Ad}_{P})$ via
$\pi\colon\thinspace P\to M$. It restricts to $G$–equivariant forms on $P$.
This yields a sequence

$\cdots\longrightarrow\Omega^{k-1}(P;\mathfrak{g})^{G}\stackrel{{\scriptstyle\tilde{d}^{A}}}{{\longrightarrow}}\Omega^{k}(P;\mathfrak{g})^{G}\stackrel{{\scriptstyle\tilde{d}^{A}}}{{\longrightarrow}}\Omega^{k+1}(P;\mathfrak{g})^{G}\longrightarrow\cdots.$

Considering the connection on the trivial bundle has the advantage of finding
a global formula for it. The curvature of the $G$–equivariant version of
$\tilde{d}^{A}$ is also often called the curvature of $A$, but it turns out to
be the pull-back of a $2$–form in $\operatorname{Ad}P$, which we used as a
definition.

######  Proposition 3.8.4.

We have the (global) formulas for $\varphi\in\Omega^{k}(P;\mathfrak{g})^{G}$:

  1. (1)

$\tilde{d}^{A}\varphi=d\varphi+[A\wedge\varphi]$;

  2. (2)

$\tilde{d}^{A}\tilde{d}^{A}\varphi=[(dA+\frac{1}{2}[A\wedge
A])\wedge\varphi]$.

###### Proof.

Let $H$ be the horizontal distribution corresponding to a connection $A$.
Recall, that given a $G$–equivariant map $P\to\mathfrak{g}$ and a vector field
$X$ on $M$, a section $\varphi$ of $\operatorname{Ad}P$ viewed as a
$G$–equivariant map $P\to\mathfrak{g}$ and a point $p\in P$, then $d^{A}$ is
given by

$(d^{A}_{X}(\varphi))(p)=d\varphi(\tilde{X}_{p}),\quad\text{where
}\tilde{X}_{p}=(d\pi|_{H_{p}})^{-1}X_{\pi(p)}\text{ is the horizontal lift of
}X_{\pi(p)}.$

Given $Z_{p}\in T_{p}P$, the connection $A$ induces the decomposition

$Z_{p}=\tilde{X}_{p}+Y^{*}_{p},\quad\text{where }Y^{*}_{p}=di_{p}Y_{e}$

for some $Y\in\mathfrak{g}$. Then given a $G$–equivariant map
$\varphi\colon\thinspace P\to\mathfrak{g}$, the a $G$–equivariant map
$\tilde{d}^{A}_{Z}(\varphi)\colon\thinspace P\to\mathfrak{g}$ given by

$\displaystyle(\tilde{d}^{A}_{Z}(\varphi))(p)$
$\displaystyle=d\varphi((d\pi_{p}|_{H_{p}})^{-1}(d\pi(Z_{p})))$
$\displaystyle=d\varphi(\tilde{X}_{p})$
$\displaystyle=d\varphi(Z_{p})-d\varphi(Y^{*}_{p})$
$\displaystyle=d\varphi(Z_{p})-d(\varphi\circ i_{p})Y_{e}.$

Notice that $\varphi\circ i_{p}(g)=\varphi(p\cdot
g)=\operatorname{Ad}_{g^{-1}}\varphi(p)$. By Theorem 2.6.5 we therefore have

$\displaystyle d(\varphi\circ i_{p})Y_{e}$
$\displaystyle=\operatorname{ad}(\varphi(p))(Y_{e})=[\varphi(p),Y_{e}]$
$\displaystyle=-[Y_{e},\varphi(p)]=-[A(Y^{*}_{p}),\varphi(p)]$
$\displaystyle=-[A(Z_{p}),\varphi(p)].$

This shows that $\tilde{d}^{A}(\varphi)=d\varphi+[A,\varphi]$ for
$\varphi\in\Omega^{0}(P;\mathfrak{g})^{G}$. Now consider
$\omega\in\Omega^{k}(P)$ for $k>0$ and
$\varphi\in\Omega^{0}(P;\mathfrak{g})^{G}$. We use induction and the Leibnitz
rule:

$\displaystyle d^{A}(\omega\otimes\varphi)$
$\displaystyle=d\omega\otimes\varphi+(-1)^{k}\omega\wedge d^{A}\varphi$
$\displaystyle=d\omega\otimes\varphi+(-1)^{k}\omega\wedge
d\varphi+(-1)^{k}\omega\wedge[A\wedge\varphi]$
$\displaystyle=d(\omega\otimes\varphi)+[A\wedge\varphi]\wedge\omega$
$\displaystyle=d(\omega\otimes\varphi)+[A\wedge(\omega\wedge\varphi)].$

This completes the computation of $\tilde{d}^{A}$. Then for
$\varphi\in\Omega^{k}(P;\mathfrak{g})^{G}$

$\displaystyle\tilde{d}^{A}\tilde{d}^{A}\varphi$
$\displaystyle=d^{2}\varphi+d[A\wedge\varphi]+[A\wedge[A\wedge\varphi]]+[A\wedge
d\varphi]$ $\displaystyle=[dA\wedge\varphi]-[A\wedge
d\varphi]-[A\wedge[A\wedge\varphi]]+[A\wedge d\varphi].$

The graded Jacobi identity gives us

$\displaystyle[A\wedge[A\wedge\varphi]]-[A\wedge[\varphi\wedge
A]]+[\varphi\wedge[A\wedge A]]$ $\displaystyle=0\quad\text{for
}\varphi\in\Omega^{\text{even}}(P;\mathfrak{g})^{G}$
$\displaystyle-[A\wedge[A\wedge\varphi]]-[A\wedge[\varphi\wedge
A]]-[\varphi\wedge[A\wedge A]]$ $\displaystyle=0\quad\text{for
}\varphi\in\Omega^{\text{odd}}(P;\mathfrak{g})^{G}.$

Therefore graded symmetry implies $2[A\wedge[A\wedge\varphi]]=[[A\wedge
A]\wedge\varphi]$ in both cases and consequently

$\tilde{d}^{A}\tilde{d}^{A}\varphi=[(dA+\tfrac{1}{2}[A\wedge
A])\wedge\varphi].\qed$

It follows immediately that $\pi^{*}(F^{A})=F^{\tilde{d}^{A}}$. We want to see
that $F^{A}\in\Omega^{2}(\operatorname{Ad}_{P})$, where

$\displaystyle\Omega^{k}(\operatorname{Ad}_{P})$
$\displaystyle\hookrightarrow\Omega^{k}(\operatorname{Hom}(\operatorname{Ad}_{P},\operatorname{Ad}_{P}))$
$\displaystyle\omega$ $\displaystyle\mapsto f\quad\text{where
}f(\varphi)=[\omega,\varphi]\text{ for
}\varphi\in\Gamma(\operatorname{Ad}_{P}).$

It as apparent from $(\tilde{d}^{A}_{Z}(\varphi))(p)=d\varphi(\tilde{X}_{p})$
for $Z_{p}=\tilde{X}_{p}+Y^{*}_{p}$, that

$(\tilde{d}^{A}_{Y^{*}}(\varphi))(p)=0.$

Therefore,

(3.8.1) $\displaystyle(dA+\tfrac{1}{2}[A\wedge A])(Z_{p},Z^{\prime}_{p})$
$\displaystyle=(dA+\tfrac{1}{2}[A\wedge
A])(\tilde{X}_{p},\tilde{X}^{\prime}_{p})$
$\displaystyle=(dA+\tfrac{1}{2}[A\wedge
A])((d\pi|_{H_{p}})^{-1}X_{\pi(p)},(d\pi|_{H_{p}})^{-1}X^{\prime}_{\pi(p)})$

Let $q\in M$ be arbitrary and $p\in\pi^{-1}(q)$ any point in the fiber over
$q$. Define the curvature $F^{A}\in\Omega^{2}(\operatorname{Ad}_{P})$ by

$F^{A}(X_{q},X^{\prime}_{q})(p):=(dA+\tfrac{1}{2}[A\wedge
A])((d\pi|_{H_{p}})^{-1}X_{q},(d\pi|_{H_{p}})^{-1}X^{\prime}_{q}).$

where we consider the above section of $\operatorname{Ad}_{P}$ as a
$G$–equivariant map $P\to\mathfrak{g}$. $F^{A}$ is well-defined, since by
$G$-invariance of the distribution $H$ we have

$d\pi|_{H_{pg}}=d\pi|_{d(r_{g})H_{p}}=d\pi|_{H_{p}}\circ(d(r_{g}))^{-1},$

and by the $G$–equivariance of $A$ we get

$\displaystyle F^{A}(X_{q},X^{\prime}_{q})(pg)$
$\displaystyle=(dA+\tfrac{1}{2}[A\wedge
A])((d\pi|_{H_{pg}})^{-1}X_{q},(d\pi|_{H_{pg}})^{-1}X^{\prime}_{q})$
$\displaystyle=(dA+\tfrac{1}{2}[A\wedge
A])((dr_{g})\circ(d\pi|_{H_{p}})^{-1}X_{q},(dr_{g})\circ(d\pi|_{H_{p}})^{-1}X^{\prime}_{q})$
$\displaystyle=\operatorname{Ad}_{g^{-1}}(dA+\tfrac{1}{2}[A\wedge
A])((d\pi|_{H_{p}})^{-1}X_{q},\circ(d\pi|_{H_{p}})^{-1}X^{\prime}_{q})$
$\displaystyle=\operatorname{Ad}_{g^{-1}}F^{A}(X_{q},X^{\prime}_{q})(p).$

This immediately gives the following proposition.

######  Proposition 3.8.5.

There is a unique 2–form $F^{A}\in\Omega^{2}(\operatorname{Ad}_{P})$ such that
for $\varphi\in\Omega^{k}(\operatorname{Ad}_{P})$

$d^{A}d^{A}\varphi=[F^{A},\varphi],$

where $\pi^{*}(F^{A})=dA+\frac{1}{2}[A\wedge
A]\in\Omega^{k}(P;\mathfrak{g})^{G}$.

The following exercise shows that the definitions of curvature of connections
in vector and principal bundles are essentially equivalent.

######  Exercise 3.8.6.

If $P\to M$ is a prinicpal bundle and $E$ is associated to $P$ via a
representation $\rho\colon\thinspace G\to\text{\it{GL}}(V)$. Show that the
$G$–equivariant linear map
$d\rho\colon\thinspace\mathfrak{g}\to\operatorname{Hom}(V,V)$ induces a map

$d\rho\colon\thinspace\Omega^{k}(\operatorname{Ad}_{P})\to\Omega^{k}(\operatorname{Hom}(E,E))$

which satisfies $d\rho(F^{A})=F^{A}_{\rho}$, where $F^{A}_{\rho}$ is the
curvature of the affine connection $d^{A}_{\rho}$ in $E$ induced by $A$.

######  Definition 3.8.7.

A connection $A$ is flat if $F^{A}=0$ pointwise. Denote by ${\mathcal{F}}_{P}$
the space of flat connections.

The above exercise shows, that for any vector bundle $E$ associated to a
principal bundle $P$ via a representation $\rho\colon\thinspace
G\to\text{\it{GL}}(V)$ the sequence

$\cdots\Omega^{k-1}(E)\stackrel{{\scriptstyle
d^{A}_{\rho}}}{{\to}}\Omega^{k}(E)\stackrel{{\scriptstyle
d^{A}_{\rho}}}{{\to}}\Omega^{k+1}(E)\to\cdots$

is a complex, if $A$ is a flat connection.

######  Proposition 3.8.8.

Let $H$ be the horizontal distribution associated to a connection $A$. Then
$H$ is involutive if and only if $A$ is flat.

###### Proof.

Let $H$ be the horizontal distribution associated to a connection $A$ and
$X,Y$ smooth vector fields in $H$. Since $A(X)=0$ and $A(Y)=0$ we have

$dA(X,Y)=XA(Y)-YA(X)-A([X,Y])=-A([X,Y]).$

It follows that

$F^{A}(d\pi(X),d\pi(Y))=dA(X,Y)+\frac{1}{2}[A,A](X,Y)=-A([X,Y]),$

which implies that $[X,Y]\in\ker A=H$ if and only if $F^{A}=0$. Therefore $H$
is involutive if and only if $A$ is flat. ∎

######  Theorem 3.8.9.

If $A$ is flat, then the holonomy representation

$\operatorname{hol}_{A}\colon\thinspace\text{Loops}(M,x_{0})\to G$

is well-defined on the fundamental group, and furthermore, the homology on the
chain complex described above is just the cohomology of $M$ with local
coefficients in $V$ given by the composite

$\pi_{1}(M,x_{0})\stackrel{{\scriptstyle\operatorname{hol}_{A}}}{{\longrightarrow}}G\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}\text{\it{GL}}(V).$

###### Proof.

Two loops $\gamma$ and $\gamma^{\prime}$ based at $x_{0}$ are homotopic, if
and only if the concatenation of the path $\gamma_{t}$ followed by
$\gamma_{-t}$ is contractible. Therefore, to see that $\operatorname{hol}_{A}$
is well-defined on $\pi_{1}(M,x_{0})$, it suffices to show that
$\operatorname{hol}_{A}$ is the identity on any contractible loop. Consider a
homotopy $h\colon\thinspace I\times I\to M$ with $h(s,0)=h(s,1)=h(1,t)=x_{0}$
from a loop to the constant loop. It follows from Frobenius’ Theorem (Theorem
2.4.10) and Proposition 3.8.8 that the horizontal distribution associated to a
flat connection has a flat chart in a neighborhood of every point $p\in P$.
Since $I\times I$ is contractible there is a flat chart in a neighborhood of
$\pi^{-1}(h(I\times I))$. In particular, this implies that the horizontal lift
of $h(0,t)$ in $M$ is homotopic to the constant path and that
$\operatorname{hol}_{A}$ is well-defined on $\pi_{1}(M)$333Note that this
argument also takes care of reparametrizations of the loops in $M$.

The statement about the homology of the complex is a generalization of de
Rham’s Theorem, that the de Rham cohomology is isomorphic to singular
cohomology, is standard. We can use sheaf theory to prove de Rham’s Theorem as
in Bredon [14] as well as the above generalization. ∎

######  Exercise 3.8.10.

Prove the Bianchi identity

$d(\pi^{*}(F^{A}))+[A\wedge\pi^{*}(F^{A})]=0.$

###  3.9. Gauge transformations

In the future it will actually make more sense to let
$F^{A}\in\Omega^{2}(P;\mathfrak{g})^{G}$.

######  Definition 3.9.1.

A principal bundle homomorphism $\Phi\colon\thinspace P\to P^{\prime}$ is a
$G$–equivariant fiber bundle homomorphism. If $P=P^{\prime}$, then $\Phi$ is
an automorphism or a gauge transformation. Denote the group of gauge
transformations by ${\mathcal{G}}_{P}$.

We identify the group of gauge transformations with the group of
$G$–equivariant maps $u\colon\thinspace P\to G$ by $u\mapsto\Phi(p)=p\cdot
u_{p}$, where $G$ acts on itself on the right by $C_{g^{-1}}$ and
$C_{g}h=ghg^{-1}$ is the conjugation action:

${\mathcal{G}}_{P}=\Gamma(P\times_{C}G).$

######  Lemma 3.9.2.

There is a natural action of ${\mathcal{G}}_{P}$ on ${\mathcal{A}}$ defined
via pull-back. In terms of $G$–equivariant maps $u\colon\thinspace P\to G$ it
takes the form

$\displaystyle{\mathcal{A}}_{P}\times{\mathcal{G}}_{P}$
$\displaystyle\to{\mathcal{A}}_{P}$ $\displaystyle(A,u)$ $\displaystyle\mapsto
A\cdot u=\operatorname{Ad}_{u^{-1}}(A)+u^{*}(\theta),$

where $\theta$ is the Maurer-Cartan form on $G$. If we identify
$\mathfrak{g}=T_{e}G$ we have have

$u^{*}(\theta)(X_{p})=(l_{u_{p}^{-1}}\circ u)_{*}(X_{p}).$

Under this action the curvature transforms as

$F^{A\cdot u}=\operatorname{Ad}_{u^{-1}}(F^{A}).$

In particular ${\mathcal{F}}_{P}$ is invariant under the action of
${\mathcal{G}}_{P}$.

###### Proof.

Certainly ${\mathcal{G}}_{P}$ acts on $\Omega^{1}(P;\mathfrak{g})$ on the
right because $\Phi^{*}\circ\Psi^{*}=(\Phi\circ\Psi)^{*}$. Note that for a
gauge transformation $\Phi$, $g\in G$ and $p\in P$ we have

$\displaystyle\Phi\circ r_{g}(p)=\Phi(pg)=\Phi(p)g=r_{g}\circ\Phi(p)$ and
$\displaystyle\Phi\circ i_{p}(g)=\Phi(pg)=\Phi(p)g=i_{\Phi(p)}g.$

Therefore for the fundamental vector field $X^{*}$ corresponding to
$X\in\mathfrak{g}$ we get

$\displaystyle(\Phi^{*}A)(X^{*}_{p})=A(\Phi_{*}(i_{p})_{*}X_{e})=A((i_{\Phi(p)})_{*}X_{e})=A(X_{\Phi(p)}^{*})=X$
and $\displaystyle
r_{g}^{*}(\Phi^{*}(A))=\Phi^{*}(r_{g}^{*}(A))=\Phi^{*}(\operatorname{Ad}_{g^{-1}}A)=\operatorname{Ad}_{g^{-1}}(\Phi^{*}(A))$

So $\Phi^{*}(A)\in{\mathcal{A}}_{P}$ for $A\in{\mathcal{A}}_{P}$. If
$\Phi(p)=p\cdot u_{p}$ for a $G$–equivariant map $u\colon\thinspace P\to G$,
then it is a simple consequence of the product formula that

$\Phi_{*}X_{p}=(r_{u_{p}})_{*}X_{p}+(i_{p})_{*}u_{*}(X_{p})$

and therefore we get

$\displaystyle(A\cdot u)(X_{p})$
$\displaystyle=A((r_{u_{p}})_{*}X_{p})+A((i_{p})_{*}u_{*}(X_{p}))$
$\displaystyle=\operatorname{Ad}_{u_{p}^{-1}}A(X_{p})+A((i_{pu_{p}})_{*}(l_{u_{p}^{-1}})_{*}u_{*}(X_{p}))$
(3.9.1)
$\displaystyle=\operatorname{Ad}_{u_{p}^{-1}}A(X_{p})+(l_{u_{p}^{-1}}\circ
u)_{*}(X_{p}))$
$\displaystyle=\operatorname{Ad}_{u_{p}^{-1}}A(X_{p})+(u^{*}\theta)(X_{p}),$

where we also used the identification $\mathfrak{g}=T_{e}G$ in line (3.9.1).
Lastly recall from Equation (3.8.1) that $F^{A}$ is zero if one of the tangent
vectors is vertical, in particular on the image of $(i_{p})_{*}$, therefore
equivariance of $A$ immediately implies

$\displaystyle F^{A\cdot u}(X_{p},Y_{p})$
$\displaystyle=F^{A}(\Phi_{*}X_{p},\Phi_{*}Y_{p})=F^{A}((r_{u_{p}})_{*}X_{p},(r_{u_{p}})_{*}Y_{p})$
$\displaystyle=dA((r_{u_{p}})_{*}X_{p},(r_{u_{p}})_{*}Y_{p})+\tfrac{1}{2}[A\wedge
A]((r_{u_{p}})_{*}X_{p},(r_{u_{p}})_{*}Y_{p})$
$\displaystyle=\operatorname{Ad}_{u_{p}^{-1}}A(X_{p},Y_{p})+\tfrac{1}{2}[(Ad_{u_{p}^{-1}}A)\wedge(Ad_{u_{p}^{-1}}A)](X_{p},Y_{p})$
$\displaystyle=Ad_{u_{p}^{-1}}F^{A}(X_{p},Y_{p}).\qed$

We will often encounter the situation where $P$ is trivializable. If we fix a
trivialization $P\cong M\times G$, then we can identify

${\mathcal{A}}_{P}\cong\Omega^{1}(M;\mathfrak{g}),\quad{\mathcal{G}}_{P}\cong
C^{\infty}(M,G).$

The action of ${\mathcal{G}}_{P}$ on ${\mathcal{A}}_{P}$ can still be written
as

$(A\cdot u)(p)=\operatorname{Ad}_{u^{-1}_{p}}A+l_{u_{p}^{-1}}u_{*}.$

###  3.10. Flat connections and the fundamental group

Fix $p_{0}\in\pi^{-1}(x_{0})$. Then a flat connection $A$ gives us a group
homomorphism

$\operatorname{hol}_{A}\colon\thinspace\pi_{1}(M,x_{0})\to G.$

In other words, we have a map
${\mathcal{F}}_{P}\to\operatorname{Hom}(\pi_{1}(M,x_{0}),G)$. Let
${\mathcal{M}}_{P}={\mathcal{F}}_{P}/{\mathcal{G}}_{P}$ be the moduli space of
flat connections in a principal $G$–bundle $P$. The following Proposition
shows that for $M$ connected

$\operatorname{hol}\colon\thinspace{\mathcal{M}}_{P}\to\operatorname{Hom}(\pi_{1}(M),G)/G$

is well-defined, particularly independent of $x_{0}$ or its lift, where $G$
acts on $\operatorname{Hom}(\pi_{1}(M),G)$ on the right via $(\rho\cdot
g)(\gamma)=g^{-1}\rho(\gamma)g$. If $G$ is a subgroup of
$\operatorname{Aut}(V)$ for some finite-dimensional vector space $V$ (or even
a Hilbert space), then $\operatorname{Hom}(\pi_{1}(M),G)$ are representations
of $\pi_{1}(M,x_{0})$.

######  Proposition 3.10.1.

Let $A$ be a flat connection in $P$, $\alpha\in\pi_{1}(M,x_{0})$ and
$p_{0}\in\pi^{-1}(x_{0})$.

  1. (1)

Let $p_{0}^{\prime}$ be another point in the fiber over $x_{0}$. Then
$p_{0}^{\prime}=p_{0}\cdot g$ and

$\operatorname{hol}_{A}(\alpha,p_{0}g)=g^{-1}\operatorname{hol}_{A}(\alpha,p_{0})g.$

  2. (2)

Let $u\colon\thinspace P\to G$ be a gauge transformation. Then we have

$\operatorname{hol}_{A\cdot
u}(\alpha,p_{0})=u_{p_{0}}^{-1}\operatorname{hol}_{A}(\alpha,p_{0})u_{p_{0}}.$

  3. (3)

Let $x_{0}^{\prime}$ be a different base point,
$p_{0}^{\prime}\in\pi^{-1}(x^{\prime}_{0})$ and $\gamma\colon\thinspace I\to
M$ a path from $x_{0}$ to $x_{0}^{\prime}$, then there exists $g\in G$ such
that

$\operatorname{hol}_{A}(\gamma*\alpha*\gamma^{-1},p_{0}^{\prime})=g\operatorname{hol}_{A}(\alpha,p_{0})g^{-1}.$

###### Proof.

Let $\beta$ be a horizontal lift of $\alpha$ with base point $p_{0}$. To show
(1) note that for the path

$\beta^{\prime}_{t}=\beta_{t}\cdot g$

for $g\in G$ we have

$A(\dot{\beta}^{\prime}_{t})=A((r_{g})_{*}\beta_{t})=\operatorname{Ad}_{g^{-1}}A(\beta_{t})=0.$

Therefore $\beta^{\prime}_{t}$ is the horizontal lift with base point
$p_{0}\cdot g$. Then

$\displaystyle\beta^{\prime}(1)$ $\displaystyle=\beta(1)g$
$\displaystyle=\beta(0)\operatorname{hol}_{A}(\alpha,p_{0})g$
$\displaystyle=\beta^{\prime}(0)g^{-1}\operatorname{hol}_{A}(\alpha,p_{0})g$

and the holonomy of $A$ along $\alpha$ with respect to the base point $p_{0}h$
is given by

$\operatorname{hol}_{A}(\alpha,p_{0}g)=g^{-1}\operatorname{hol}_{A}(\alpha,p_{0})g.$

To show (2) let $\beta^{\prime}(t)=\beta(t)w_{\beta(t)}$ where $w=u^{-1}$.
Then the product rule implies

$\dot{\beta}^{\prime}_{t}=(i_{\beta(t)})_{*}w_{*}\dot{\beta}_{t}+(r_{w(\beta_{t})})_{*}\dot{\beta}_{t}.$

If $B$ is any connection then

$\displaystyle B(\dot{\beta}^{\prime}_{t})$
$\displaystyle=B((i_{\beta(t)w_{\beta(t)}})_{*}(l_{w^{-1}_{\beta(t)}})_{*}w_{*}\dot{\beta}_{t})+B((r_{w(\beta_{t})})_{*}\dot{\beta}_{t})$
$\displaystyle=(l_{w^{-1}_{\beta(t)}})_{*}w_{*}\dot{\beta}_{t}+\operatorname{Ad}_{w_{\beta(t)}^{-1}}B(\dot{\beta}_{t})$
$\displaystyle=w^{*}(\theta)(\dot{\beta}_{t})+\operatorname{Ad}_{w_{\beta(t)}^{-1}}B(\dot{\beta}_{t})$

as well as

$(B\cdot
w)(\dot{\beta}_{t})=\operatorname{Ad}_{w_{\beta(t)}^{-1}}B(\dot{\beta}_{t})+w^{*}(\theta)(\dot{\beta}_{t}).$

Therefore we get for $B=A\cdot u$

$A(\dot{\beta}_{t})=(B\cdot
w)(\dot{\beta}_{t})=B(\dot{\beta}^{\prime}_{t})=(A\cdot
u)(\dot{\beta}^{\prime}_{t}).$

Since $\beta$ is the horizontal lift of $\alpha$ with respect to $A$ starting
at $p_{0}$, $\beta^{\prime}$ is the horizontal lift of $\alpha$ with respect
to $A\cdot u$ starting at $\beta^{\prime}_{0}=p_{0}u^{-1}_{p_{0}}$. Then

$u_{\beta(1)}=u_{\beta(0)\cdot\operatorname{hol}_{A}(\alpha,p_{0})}=C_{(\operatorname{hol}_{A}(\alpha,p_{0}))^{-1}}u_{p_{0}}$

implies

$\displaystyle\beta^{\prime}(1)$ $\displaystyle=\beta(1)\cdot
u_{\beta(1)}^{-1}$
$\displaystyle=\beta(0)\cdot\operatorname{hol}_{A}(\alpha,p_{0})\cdot
C_{(\operatorname{hol}_{A}(\alpha,p_{0}))^{-1}}u_{p_{0}}^{-1}$
$\displaystyle=\beta^{\prime}(0)\cdot\operatorname{hol}_{A}(\alpha,p_{0}).$

Therefore $\operatorname{hol}_{A\cdot
u}(\alpha,p_{0}u^{-1}_{p_{0}})=\operatorname{hol}_{A}(\alpha,p_{0})$ and by
(1)

$\displaystyle\operatorname{hol}_{A\cdot u}(\alpha,p_{0})$
$\displaystyle=u^{-1}_{p_{0}}\operatorname{hol}_{A\cdot
u}(\alpha,p_{0}u^{-1}_{p_{0}})u_{p_{0}}$
$\displaystyle=u_{p_{0}}^{-1}\operatorname{hol}_{A}(\alpha,p_{0})u_{p_{0}}.$

The horizontal lift $\beta_{\gamma}$ of $\gamma$ starting at $\beta(0)=p_{0}$
ends at $\beta_{\gamma}(1)=p_{0}^{\prime}g$ for some $g\in G$. We know from
the proof of (1) that any other horizontal lift is of the form
$\beta_{\gamma}\cdot g^{\prime}$ for some $g^{\prime}\in G$. In particular the
horizontal lift $\beta_{\gamma}\cdot\operatorname{hol}_{A}(\alpha,p_{0})$
starts at $\beta(1)$. This shows that
$(\beta_{\gamma}\cdot\operatorname{hol}_{A}(\alpha,p_{0}))*\beta*\beta_{\gamma}^{-1}$
is a horizontal lift of $\gamma*\alpha*\gamma^{-1}$ based at $p_{0}^{\prime}$.

$\displaystyle((\beta_{\gamma}\cdot\operatorname{hol}_{A}(\alpha,p_{0}))*\beta*\beta_{\gamma}^{-1})(1)$
$\displaystyle=\beta_{\gamma}(1)\cdot\operatorname{hol}_{A}(\alpha,p_{0})$
$\displaystyle=\beta_{\gamma}^{-1}(0)\cdot\operatorname{hol}_{A}(\alpha,p_{0})$
$\displaystyle=((\beta_{\gamma}\cdot\operatorname{hol}_{A}(\alpha,p_{0}))*\beta*\beta_{\gamma}^{-1})(0)\cdot\operatorname{hol}_{A}(\alpha,p_{0})$

Therefore by (1)

$\operatorname{hol}_{A}(\gamma*\alpha*\gamma^{-1},p_{0}^{\prime})=g\operatorname{hol}_{A}(\gamma*\alpha*\gamma^{-1},p_{0}^{\prime}\cdot
g)g^{-1}=g\operatorname{hol}_{A}(\alpha,p_{0})g^{-1}.\qed$

We have seen how to get a homomorphism
$\rho\colon\thinspace\pi_{1}(M,x_{0})\to G$ from a flat connection $A$ in a
principal $G$–bundle $P$. To associate a principal $G$–bundle to a
homomorphism $\rho\colon\thinspace\pi_{1}(M,x_{0})\to G$ with $x_{0}\in M$,
let $\widetilde{M}$ be the universal cover of $M$. Fix a base point
$p_{0}\in\widetilde{M}$ and identify $\widetilde{M}$ with the space of
homotopy classes of paths in $M$ starting at $x_{0}$. Then $\pi_{1}(M,x_{0})$
naturally acts on $\widetilde{M}$ from the right. For
$\rho\colon\thinspace\pi_{1}(M,x_{0})\to G$ we define the principal $G$–bundle

$P_{\rho}:=\widetilde{M}\times_{\rho}G=(\widetilde{M}\times G)/\sim$

where

$(\tilde{x},g)\sim(\tilde{x}\cdot\alpha,\rho(\alpha)^{-1}g),\quad(\tilde{x},g)\in\widetilde{M}\times
G,\alpha\in\pi_{1}(M).$

Pulling back the Maurer-Cartan form $\theta\in\Omega(G;\mathfrak{g})$ to
$\widetilde{M}\times G$ defines a natural flat connection
$\tilde{A}=\pi_{G}^{*}(\theta)$ on $\widetilde{M}\times G$ since

$\displaystyle\tilde{A}(X^{*}_{(\tilde{x},g)})=\tilde{A}((i_{(\tilde{x},g)})_{*}(X_{e}))=\theta((l_{g})_{*}X_{e})=X$
$\displaystyle\quad\text{for }X\in\mathfrak{g},$ $\displaystyle
r^{*}_{h}\tilde{A}=\pi^{*}_{G}r^{*}_{h}\theta=\pi^{*}_{G}\operatorname{Ad}_{h^{-1}}\theta=\operatorname{Ad}_{h^{-1}}\tilde{A}$
by Exercise 3.7.2 and $\displaystyle
d\tilde{A}+\tfrac{1}{2}[\tilde{A}\wedge\tilde{A}]=\pi^{*}_{G}(d\theta+\tfrac{1}{2}[\theta,\theta])=0$
$\displaystyle\quad\text{by Proposition \ref{PropMaurerCartan}}.$

This also induces a flat connection $A_{\rho}$ on $P_{\rho}$ because