Differential geometry and Lie groups. A second course

580 Pages·2019·2.438 MB·English

by  Gallier J., Quaintance J

Differential Geometry and Lie Groups
A Second Course
Jean Gallier and Jocelyn Quaintance
Department of Computer and Information Science
University of Pennsylvania
Philadelphia, PA 19104, USA
e-mail: [email protected]
(cid:13)c Jean Gallier
Please, do not reproduce without permission of the authors
October 11, 2019
2
To my daughter Mia, my wife Anne,
my son Philippe, and my daughter Sylvie.
To my parents Howard and Jane.
Preface
This book is written for a wide audience ranging from upper undergraduate to advanced
graduatestudentsinmathematics,physics,andmorebroadlyengineeringstudents,especially
in computer science. Basically, it covers topics which belong to a second course in differential
geometry. The reader is expected to be familiar with the theory of manifolds and with some
elements of Riemannian geometry, including connections, geodesics, and curvature. Some
familiarity with the material presented in the following books is more than sufficient: Tu
[105] (the first three chapters), Warner [109] (the first chapter and parts of Chapters 2, 3, 4),
Do Carmo [37] (the first four chapters), Gallot, Hulin, Lafontaine [48] (the first two chapters
and parts of Chapter 3), O’Neill [84] (Chapters 1 and 3), and Gallier and Quaintance [47],
which contains all the preliminaries needed to read this book.
The goal of differential geometry is to study the geometry and the topology of manifolds
using techniques involving differentiation in one way or another. The pilars of differential
geometry are:
(1) Riemannian metrics.
(2) Connections.
(3) Geodesics.
(4) Curvature.
There are many good books covering the above topics, and we also provided our own account
(Gallier and Quaintance [47]). One of the goals of differential geometry is also to be able to
generalize “calculus on Rn” to spaces more general than Rn, namely manifolds. We would
like to differentiate functions f: M → R defined on a manifold, optimize functions (find
their minima or maxima), and also to integrate such functions, as well as compute areas and
volumes of subspaces of our manifold.
The generalization of the notion of derivative of a function defined on a manifold is the
notion of tangent map, and the notions of gradient and Hessian are easily generalized to
manifolds equipped with a connection (or a Riemannian metric, which yields the Levi-Civita
connection). However, the problem of defining the integral of a function whose domain is a
manifold remains.
3
4
One of the main discoveries made at the beginning of the twentieth century by Poincar´e
´
and Elie Cartan, is that the “right” approach to integration is to integrate differential forms,
and not functions. To integrate a function f, we integrate the form fω, where ω is a volume
form on the manifold M. The formalism of differential forms takes care of the process of
the change of variables quite automatically, and allows for a very clean statement of Stokes’
theorem.
Differential forms can be combined using a notion of product called the wedge product,
but what really gives power to the formalism of differential forms is the magical operation d
of exterior differentiation. Given a form ω, we obtain another form dω, and remarkably, the
following equation holds
ddω = 0.
As silly as it looks, the above equation lies at the core of the notion of cohomology, a
powerful algebraic tool to understanding the topology of manifolds, and more generally of
topological spaces.
´
Elie Cartan had many of the intuitions that led to the cohomology of differential forms,
but it was Georges de Rham who defined it rigorously and proved some important theorems
about it. It turns out that the notion of Laplacian can also be defined on differential forms
using a device due to Hodge, and some important theorems can be obtained: the Hodge
decomposition theorem, and Hodge’s theorem about the isomorphism between the de Rham
cohomology groups and the spaces of harmonic forms. Differential forms can also be used to
define the notion of curvature of a connection on a certain type of manifold called a vector
bundle.
The theory of differential forms is one of the main tools in geometry and topology. This
theory has a surprisingly large range of applications, and it also provides a relatively easy
access to more advanced theories such as cohomology. For all these reasons, it is really an
indispensable theory, and anyone with more than a passible interest in geometry should be
familiar with it.
´
The theory of differential forms was initiated by Poincar´e and further elaborated by Elie
Cartan at the end of the nineteenth century. Differential forms have two main roles:
(1) Describe various systems of partial differential equations on manifolds.
(2) To define various geometric invariants reflecting the global structure of manifolds or
bundles. Such invariants are obtained by integrating certain differential forms.
In this book, we discuss the following topics.
(1) Differential forms, including vector-valued differential forms and differential forms on
Lie groups.
(2) An introduction to de Rham cohomology.
5
(3) Distributions and the Frobenius theorem.
(4) Integration on manifolds, starting with orientability, volume forms, and ending with
Stokes’ theorem on regular domains.
(5) Integration on Lie groups.
(6) Spherical harmonics and an introduction to the representations of compact Lie groups.
(7) Operators on Riemannian manifolds: Hodge Laplacian, Laplace–Beltrami Laplacian,
and Bochner Laplacian.
(8) Fibre bundles, vector bundles, principal bundles, and metrics on bundles.
(9) Connections and curvature in vector bundles, culminating with an introduction to
Pontrjagin classes, Chern classes, and the Euler class.
(10) Clifford algebras, Clifford groups, and the groups Pin(n), Spin(n), Pin(p,q) and
Spin(p,q).
Topics (3)-(7) have more of an analytic than a geometric flavor. Topics (8) and (9)
belong to the core of a second course on differential geometry. Clifford algebras and Clifford
groups constitute a more algebraic topic. These can be viewed as a generalization of the
quaternions. The groups Spin(n) are important because they are the universal covers of the
groups SO(n).
Differential forms constitute the main tool needed to understand and prove many of the
resultspresentedinthisbook. Thusoneneedhaveasolidunderstandingofdifferentialforms,
which turn out to be certain kinds of skew-symmetric (also called alternating) tensors.
Consequently, we made the (perhaps painful) decision to provide a fairly detailed ex-
position of tensors, starting with arbitrary tensors, and then specializing to symmetric and
alternatingtensors. Inparticular, weexplainrathercarefullytheprocessoftakingthedualof
a tensor (of all three flavors). Tensors, symmetric tensors, tensor algebras, and symmetrc al-
gebras, are discussed on Chapter 1. Alternating tensors, and exterior algebras, are discussed
in Chapter 2. The Hodge ∗ operator is introduced, we discuss criteria for the decomposablity
of an alternating tensor in terms of hook operators, and present the Grassmann-Plu¨cker’s
equations.
Chapter 3 is devoted to a thorough presentation of differential forms, including vector-
valued differential forms, differential forms on Lie Groups, and Maurer-Cartan forms. We
also introduce de Rham cohomology.
Chapter 4 is a short chapter devoted to distributions and the Frobenius theorem. Distri-
butionsareageneralizationofvectorfields,andtheissueistounderstandwhenadistribution
is integrable. The Frobenius theorem gives a necessary and sufficient condition for a distri-
bution to have an integral manifold at every point. One version of the Frobenius theorem is
stated in terms of vector fields, the second version in terms of differential forms.
6
The theory of integration on manifolds and Lie groups is presented in Chapter 5. We
introducethenotionoforientationofasmoothmanifold(ofdimensionn), volumeforms, and
then explain how to integrate a smooth n-form with compact support. We define densities
which allow integrating n-forms even if the manifold is not orientable, but we do not go into
the details of this theory. We define manifolds with boundary, and explain how to integrate
forms on certain kinds of manifolds with boundaries called regular domains. We state and
prove a version of the famous result known as Stokes’ theorem. In the last section we discuss
integrating functions on Riemannian manifolds or Lie groups.
The main theme of Chapter 6 is to generalize Fourier analysis on the circle to higher
dimensional spheres. One of our goals is to understand the structure of the space L2(Sn)
of real-valued square integrable functions on the sphere Sn, and its complex analog L2(Sn).
C
Both are Hilbert spaces if we equip them with suitable inner products. It turns out that each
of L2(Sn) and L2(Sn) contains a countable family of very nice finite dimensional subspaces
C
H (Sn) (and HC(Sn)), where H (Sn) is the space of (real) spherical harmonics on Sn, that
k k k
is, the restrictions of the harmonic homogeneous polynomials of degree k (in n + 1 real
variables) to Sn (and similarly for HC(Sn)); these polynomials satisfy the Laplace equation
k
∆P = 0,
where the operator ∆ is the (Euclidean) Laplacian,
∂2 ∂2
∆ = +···+ .
∂x2 ∂x2
1 n+1
Remarkably, each space H (Sn) (resp. HC(Sn)) is the eigenspace of the Laplace-Beltrami
k k
operator ∆ on Sn, a generalization to Riemannian manifolds of the standard Laplacian
Sn
(in fact, H (Sn) is the eigenspace for the eigenvalue −k(n + k − 1)). As a consequence,
k
the spaces H (Sn) (resp. HC(Sn)) are pairwise orthogonal. Furthermore (and this is where
k k
analysis comes in), the set of all finite linear combinations of elements in (cid:83)∞ H (Sn) (resp.
k=0 k
(cid:83)∞ HC(Sn)) is is dense in L2(Sn) (resp. dense in L2(Sn)). These two facts imply the
k=0 k C
following fundamental result about the structure of the spaces L2(Sn) and L2(Sn).
C
The family of spaces H (Sn) (resp. HC(Sn)) yields a Hilbert space direct sum decompo-
k k
sition
∞ ∞
(cid:77) (cid:77)
L2(Sn) = H (Sn) (resp. L2(Sn) = HC(Sn)),
k C k
k=0 k=0
which means that the summands are closed, pairwise orthogonal, and that every f ∈ L2(Sn)
(resp. f ∈ L2(Sn)) is the sum of a converging series
C
∞
(cid:88)
f = f
k
k=0
7
in the L2-norm, where the f ∈ H (Sn) (resp. f ∈ HC(Sn)) are uniquely determined
k k k k
functions. Furthermore, given any orthonormal basis (Y1,...,Yak,n+1) of H (Sn), we have
k k k
a
k,n+1
(cid:88)
f = c Ymk, with c = (cid:104)f,Ymk(cid:105) .
k k,mk k k,mk k Sn
m =1
k
The coefficients c are “generalized” Fourier coefficients with respect to the Hilbert
k,m
k
basis {Ymk | 1 ≤ m ≤ a , k ≥ 0}; see Theorems 6.18 and 6.19.
k k k,n+1
When n = 2, the functions Ymk correspond to the spherical harmonics, which are defined
k
intermsoftheLegendrefunctions. Alongtheway, weprovethefamousFunk–Heckeformula.
The purpose of Section 6.9 is to generalize the results about the structure of the space
of functions L2(Sn) defined on the sphere Sn, especially the results of Sections 6.5 and 6.6
C
(such as Theorem 6.19, except part (3)), to homogeneous spaces G/K where G is a compact
Lie group and K is a closed subgroup of G.
The first step is to consider the Hilbert space L2(G) where G is a compact Lie group
C
and to find a Hilbert sum decomposition of this space. The key to this generalization is the
notion of (unitary) linear representation of the group G.
The result that we are alluding to is a famous theorem known as the Peter–Weyl theorem
about unitary representations of compact Lie groups.
The Peter–Weyl theorem can be generalized to any representation V : G → Aut(E) of G
into a separable Hilbert space E, and we obtain a Hilbert sum decomposition of E in terms
of subspaces E of E.
ρ
The next step is to consider the subspace L2(G/K) of L2(G) consisting of the functions
C C
that are right-invariant under the action of K. These can be viewed as functions on the
homogeneous space G/K. Again we obtain a Hilbert sum decomposition. It is also interest-
ing to consider the subspace L2(K\G/K) of functions in L2(G) consisting of the functions
C C
that are both left and right-invariant under the action of K. The functions in L2(K\G/K)
C
can be viewed as functions on the homogeneous space G/K that are invariant under the left
action of K.
Convolution makes the space L2(G) into a non-commutative algebra. Remarkably, it is
C
possible to characterize when L2(K\G/K) is commutative (under convolution) in terms of
C
a simple criterion about the irreducible representations of G. In this situation, (G,K) is a
called a Gelfand pair.
When (G,K) is a Gelfand pair, it is possible to define a well-behaved notion of Fourier
transform on L2(K\G/K). Gelfand pairs and the Fourier transform are briefly considered
C
in Section 6.11.
Chapter 7 deals with various generalizations of the Lapacian to manifolds.
The Laplacian is a very important operator because it shows up in many of the equations
used in physics to describe natural phenomena such as heat diffusion or wave propagation.
8
Therefore,itishighlydesirabletogeneralizetheLaplaciantofunctionsdefinedonamanifold.
´
Furthermore, in the late 1930’s, Georges de Rham (inspired by Elie Cartan) realized that it
was fruitful to define a version of the Laplacian operating on differential forms, because of
a fundamental and almost miraculous relationship between harmonics forms (those in the
kernel of the Laplacian) and the de Rham cohomology groups on a (compact, orientable)
smooth manifold. Indeed, as we will see in Section 7.6, for every cohomology group Hk (M),
DR
every cohomology class [ω] ∈ Hk (M) is represented by a unique harmonic k-form ω; this is
DR
the Hodge theorem. The connection between analysis and topology lies deep and has many
important consequences. For example, Poincar´e duality follows as an “easy” consequence of
the Hodge theorem.
Technically, the Hodge Laplacian can be defined on differential forms using the Hodge ∗
operator (Section 2.5). On functions, there is an alternate and equivalent definition of the
Laplacian using only the covariant derivative and obtained by generalizing the notions of
gradient and divergence to functions on manifolds.
Another version of the Laplacian on k-forms can be defined in terms of a generalization
of the Levi-Civita connection ∇: X(M)×X(M) → X(M) to k-forms viewed as a linear map
∇: Ak(M) → Hom (X(M),Ak(M)),
C∞(M)
and in terms of a certain adjoint ∇∗ of ∇, a linear map
∇∗: Hom (X(M),Ak(M)) → Ak(M).
C∞(M)
We obtain the Bochner Laplacian (or connection Laplacian ) ∇∗∇. Then it is natural to
wonder how the Hodge Laplacian ∆ differs from the connection Laplacian ∇∗∇?
Remarkably, there is a formula known as Weitzenb¨ock’s formula (or Bochner’s formula)
of the form
∆ = ∇∗∇+C(R ),
∇
where C(R ) is a contraction of a version of the curvature tensor on differential forms (a
∇
fairly complicated term). In the case of one-forms,
∆ = ∇∗∇+Ric,
where Ric is a suitable version of the Ricci curvature operating on one-forms.
Weitzenbo¨ck-type formulae are at the root of the so-called “Bochner technique,” which
consists in exploiting curvature information to deduce topological information.
Chapter 8 is an introduction to bundle theory; we discuss fibre bundles, vector bundles,
and principal bundles.
Intuitively, a fibre bundle over B is a family E = (E ) of spaces E (fibres) indexed
b b∈B b
by B and varying smoothly as b moves in B, such that every E is diffeomorphic to some
b
9
prespecified space F. The space E is called the total space, B the base space, and F the
fibre. A way to define such a family is to specify a surjective map π: E → B. We will assume
that E, B, F are smooth manifolds and that π is a smooth map. The type of bundles that
we just described is too general and to develop a useful theory it is necessary to assume that
locally, a bundle looks likes a product. Technically, this is achieved by assuming that there
is some open cover U = (U ) of B and that there is a family (ϕ ) of diffeomorphisms
α α∈I α α∈I
ϕ : π−1(U ) → U ×F.
α α α
Intuitively, above U , the open subset π−1(U ) looks like a product. The maps ϕ are called
α α α
local trivializations.
The last important ingredient in the notion of a fibre bundle is the specifiction of the
“twisting” of the bundle; that is, how the fibre E = π−1(b) gets twisted as b moves in the
b
base space B. Technically, such twisting manifests itself on overlaps U ∩U (cid:54)= ∅. It turns
α β
out that we can write
ϕ ◦ϕ−1(b,x) = (b,g (b)(x))
α β αβ
for all b ∈ U ∩U and all x ∈ F. The term g (b) is a diffeomorphism of F. Then we require
α β αβ
that the family of diffeomorphisms g (b) belongs to a Lie group G, which is expressed by
αβ
specifying that the maps g , called transitions maps, are maps
αβ
g : U ∩U → G.
αβ α β
The purpose of the group G, called the structure group, is to specify the “twisting” of the
bundle.
Fibre bundles are defined in Section 8.1. The family of transition maps g satisfies an
αβ
important condition on nonempty overlaps U ∩U ∩U called the cocycle condition:
α β γ
g (b)g (b) = g (b)
αβ βγ αγ
(where g (b),g (b),g (b) ∈ G), for all α,β,γ such that U ∩ U ∩ U (cid:54)= ∅ and all b ∈
αβ βγ αγ α β γ
U ∩U ∩U .
α β γ
In Section 8.2 we define bundle morphisms, and the notion of equivalence of bundles over
the same base, following Hirzebruch [57] and Chern [22]. We show that two bundles (over
the same base) are equivalent if and only if they are isomorphic.
In Section 8.3 we describe the construction of a fibre bundle with prescribed fibre F and
structure group G from a base manifold, B, an open cover U = (U ) of B, and a family of
α α∈I
maps g : U ∩U → G satisfying the cocycle condition, called a cocycle. This construction
αβ α β
is the basic tool for constructing new bundles from old ones.
Section 8.4 is devoted to a special kind of fibre bundle called vector bundles. A vector
bundle is a fibre bundle for which the fibre is a finite-dimensional vector space V, and the
structure group is a subgroup of the group of linear isomorphisms (GL(n,R) or GL(n,C),
10
where n = dimV). Typical examples of vector bundles are the tangent bundle TM and the
cotangent bundle T∗M of a manifold M. We define maps of vector bundles, and equivalence
of vector bundles.
In Section 8.5 we describe various operations on vector bundles: Whitney sums, ten-
sor products, tensor powers, exterior powers, symmetric powers, dual bundles, and Hom
bundles. We also define the complexification of a real vector bundle.
In Section 8.6 we discuss properties of the sections of a vector bundle ξ. We prove that
the space of sections Γ(ξ) is finitely generated projective C∞(B)-module.
Section 8.7 is devoted to the the covariant derivative of tensor fields, and to the duality
between vector fields and differential forms.
In Section 8.8 we explain how to give a vector bundle a Riemannian metric. This is
achieved by supplying a smooth family ((cid:104)−,−(cid:105) ) of inner products on each fibre π−1(b)
b b∈B
aboveb ∈ B. Wedescribethenotionofreductionofthestructuregroupanddefineorientable
vector bundles.
In Section 8.9 we consider the special case of fibre bundles for which the fibre coincides
with the structure group G, which acts on itself by left translations. Such fibre bundles are
called principal bundles. It turns out that a principal bundle can be defined in terms of a free
right action of Lie group on a smooth manifold. When principal bundles are defined in terms
of free right actions, the notion of bundle morphism is also defined in terms of equivariant
maps.
There are two constructions that allow us to reduce the study of fibre bundles to the
study of principal bundles. Given a fibre bundle ξ with fibre F, we can construct a principal
bundle P(ξ) obtained by replacing the fibre F by the group G. Conversely, given a principal
bundle ξ and an effective action of G on a manifold F, we can construct the fibre bundle
ξ[F] obtained by replacing G by F. The maps
ξ (cid:55)→ ξ[F] and ξ (cid:55)→ P(ξ)
induce a bijection between equivalence classes of principal G-bundles and fibre bundles (with
structure group G). Furthermore, ξ is a trivial bundle iff P(ξ) is a trivial bundle.
Section 8.10 is devoted to principal bundles that arise from proper and free actions of a
Lie group. When the base space is a homogenous space, which means that it arises from a
transitive action of a Lie group, then the total space is a principal bundle. There are many
illustrations of this situation involving SO(n+1) and SU(n+1).
In Chapter 9, we discuss connections and curvature in vector bundles. In Section 9.2 we
define connections on a vector bundle. This can be done in two equivalent ways. One of the
two definitions is more abstract than the other because it involves a tensor product, but it
is technically more convenient. This definition states that a connection on a vector bundle
ξ, as an R-linear map
∇: Γ(ξ) → A1(B)⊗ Γ(ξ) (∗)
C∞(B)