Differential Geometry
Michaelmas 2006
Dr. G.P. Paternain∗
Chris Almost†
See Dr. Paternain’s website for the room and time of the next examples class
(at the beginning of Lent term).
Contents
Contents
1
1 Manifolds
1.1 Definition and First Examples .
1.2 Tangent Space and Differentials
1.3 Tangent Bundles . . . . . . . . .
1.4 Submanifolds . . . . . . . . . . .
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2
2
5
8
11
2 Forms and Bundles
2.1 Differential Forms . . . . . . .
2.2 Orientability and Integration
2.3 Metrics . . . . . . . . . . . . .
2.4 Bundles . . . . . . . . . . . . .
2.5 Connections . . . . . . . . . .
2.6 Curvature . . . . . . . . . . . .
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13
13
16
21
23
27
32
3 Riemannian Metrics
3.1 Metric Connections . . . . . . . . . . . .
3.2 Levi-Civita connection . . . . . . . . . .
3.3 Curvature revisited . . . . . . . . . . . .
3.4 Sectional, Ricci, and Scalar curvature
3.5 Laplace(-Bertrami) operator . . . . . .
3.6 Yang-Mills Equations . . . . . . . . . . .
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33
33
34
35
37
37
43
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Index
∗
†
47
[email protected]
[email protected]
1
2
Differential Geometry
Notes: (available on the internet)
(i) Alexei Kovalev (followed most closely)
(ii) Mihalis Dafermos (these have an emphasis on calculus of variations)
Book:
(i) J. Jost, Riemannian Geometry & Geometric Analysis, Springer, Universitext
(chapters 1, 2, 3)
1
Manifolds
1.1
Definition and First Examples
1.1.1 Definition. A topological space M is a set with a specified class of open sets,
such that
(i) ∅ and M are open;
(ii) the intersection of two open sets is open;
(iii) an arbitrary union of open sets is open.
M is Haudorff if given p1 , p2 ∈ M there are open sets Ui (i = 1, 2) such that pi ∈ Ui
and U1 ∩ U2 = ∅. M is second countable if one can find a coutable collection B
of open sets of M such that any open set U ⊆ M can be written as a union of
elements in B.
1.1.2 Definition. Recall that a continuous map is a map such that the preimage
of every open set is open. A homeomorphism is a continuous bijection with a
continuous inverse.
1.1.3 Definition. A homeomorphism ϕ : U → V , where U ⊆ M is open and
V ⊆ Rd is open is called a chart and U the coordinate neighbourhood.
Insert standard chart picture here with change of local coordinates.
Suppose you have charts ϕ : U → V and ψ : U 0 → V 0 . Then ϕ ◦ ψ−1 is a map
between subsets of Rd .
1.1.4 Definition. A C ∞ differentiable structure (or a smooth structure) on M is a
collection of coordinate charts ϕα : Uα → Vα ⊆ Rd (same d for all α) such that
S
(i) α Uα = M ;
(ii) any two charts are compatible, namely for all α, β the change of local coordinates ϕβ ◦ ϕα−1 is C ∞ on its domain ϕα (Uβ ∩ Uα ) (this is equivalent to
requiring that it has continuous partial derivatives of all orders);
Definition and First Examples
3
(iii) the collection of charts {(Uα , ϕα )} is maximal with respect to property (ii),
namely if a chart (U, ϕ) is compatible with (Uα , ϕα ) for all α then (U, ϕ) is
included in the collection.
In this case d is the dimension of M , denoted dim M .
Remark.
(i) The change of local coordinates is a diffeomorphism, a C ∞ map with C ∞
inverse.
(ii) We only really need to worry about (i) and (ii), since there is a unique way
of extending a collection of charts satisfying (i) and (ii) to a maximal one:
just add all the compatible charts (proof is a Zorn’s Lemma arugment).
(iii) If we start just with bijective charts, we can mount a topology on M by
defining D ⊆ M to be open if and only if ϕ(D ∩ U) is open in Rd for every
chart ϕ : U → V ⊆ Rd . This is the topology induced by the C ∞ structure.
1.1.5 Examples.
(i) Rd is a manifold, as is any open subset of Rd .
Pn
(ii) S n = {(x 0 , . . . , x n ) ∈ Rn+1 | i=0 x i2 = 1}, with smooth structure given by
stereographic projection from the north and south poles. This is the canonical smooth structure on S n .
(iii) Real projective space RPn = (Rn+1 \ {0})/R∗ , with smooth structure given
by the charts (0 ≤ i ≤ n)
ϕi : Ui = {[x 0 : · · · : x n ] | x i 6= 0} → Rn
: [x 0 : · · · : x n ] 7→
x1
xi
, . . . , î, . . . ,
xn
xi
omitting the i th component. It is clear that these charts are compatible.
Check that RPn is a compact manifold. (Hint: there is a continuous projection S n → RPn .)
In the 50’s J. Milnor showed that S 7 has many different smooth structures.
He won a Field’s Medal for his work. In the 80’s an even more startling fact was
discovered: R4 has uncountably many smooth structures! The Smooth Poincaré
Conjecture asks whether S 4 has an exoctic (non-canonical) smooth structure.
1.1.6 Definition. Let M and N be smooth manifolds and f : M → N be continuous. We say that f is a smooth map if for all p ∈ M and charts (U, ϕ) containing p
and (V, ψ) containing f (p) the map ψ ◦ f ◦ ϕ −1 is C ∞ on its domain of definition.
1.1.7 Definition. A continuous map f : M → N is a diffeomorphism if f is a
smooth bijection with smooth inverse. In this case we say that M and N are
diffeomorphic
4
Differential Geometry
1.1.8 Exercise. Show that if M and N are manifolds then M × N is a manifold in
a canonical way.
1.1.9 Example (Lie groups). A Lie group is a group G which is also a smooth
manifold such that the map G × G → G : (σ, τ) 7→ στ−1 is a smooth map.
2
(i) G L(n, R) is a Lie group since it embeds naturally in Rn as an open set and
matrix multiplication is given by polynomials in the coefficients.
(ii) O(n) = {A ∈ G L(n, R) | AAT = I} is a Lie group, as we shall see below.
1.1.10 Exponential and Logarithm of Matrices. Let A be an n × n real matrix.
The exponential of A is defined to be
1
1
1
exp(A) = I + A + A2 + A3 + · · · + An + · · · .
2
6
n!
It can be shown, using the Weierstrass M -test, that this series is uniformly convergent on compact subsets of M (n, R). In fact the map A 7→ exp(A) is C ∞ . The
exponential map has the following properties.
(i) exp(AT ) = (exp(A)) T ;
(ii) exp(C −1 AC) = C −1 exp(A)C;
(iii) Warning: it does not hold in general that exp(A + B) = exp(A) exp(B), but it
does hold if A and B commute. In particular, exp(A) exp(−A) = I.
Similarily, we define the logarithm of a matrix A to be
1
(−1)n+1 n
1
A + ··· .
log(A) = A − A2 + A3 + · · · +
2
3
n
This series also converges uniformly on compact sets, provided that |A − I| < 1,
and A 7→ log(A) is C ∞ for all A with |A − I| < 1. We have
(i) exp(log(A)) = A if |A − I| < 1;
(ii) log(exp(A)) = A if |A| < log 2.
Take A such that |A − I| < 1. If A ∈ O(n) then log(A) ∈ S, the set of skewsymmetric matrices. Indeed, AA = I implies that
exp(log(A)) exp((log(A)) T ) = I
by (i), so
exp((log(A)) T ) = exp(− log(A)).
But | log(A)| ≤ log |A| < 2 since |A − I| < 1, so by (ii)
(log(A)) T = − log(A).
Tangent Space and Differentials
5
The dimension of the space of skew-symmetric matrices is 21 n(n − 1). Consider
1
ϕ:U →S∼
= R 2 n(n−1) : A 7→ log(A),
where U = {A ∈ O(n) | |A − I| < 1}. Then ϕ is a chart around the identity. For any
C ∈ O(n), let ϕC (A) = log(C −1 A) for A in a small open set around C. It remains
to check that change of coordinates is a smooth map, but this follows from the
definition of Lie group (in particular that multiplication is smooth).
1.2
Tangent Space and Differentials
The tangent space Tp M
For each point p ∈ M we would like to define the space of tangent vectors to M
at p, which we will denote Tp M . It should be a vector space of dimension dim M ,
and whenever f : M → N is a smooth map, there should be an associated linear
map d f p : Tp M → T f (p) N that satisfies the chain rule, which will be called the
differential of f
Intuitively, any tangent vector to a curve α : I → M , where 0 ∈ I ⊆ R is
an open interval and α(0) = p, is a tangent vector to M at p, and any tangent
vector should arise in this way. Let us first consider the simplest case, where the
manifold under consideration is an open set U ⊆ Rn . Let p ∈ U and consider
all smooth curves through p (i.e. all smooth curves α : I → U, where 0 ∈ I ⊆ R
is an open interval and α(0) = p). We say that two such curves α1 and α2 are
equivalent if α01 (0) = α02 (0), since all we really care about is the tangent vector at
p. Let Tp U be the set of equivalence classes. There is a natural bijection between
Tp U → Rn given by [α] 7→ α0 (0), since each equivalence class is uniquely defined
by its tangent vector at p. If f : U → Rm is a smooth map then, from multivariable
calculus,
d f (α(t)) = d f p (α0 (0)),
d t t=0
∂f
where d f p = ( ∂ xi )i j is the usual differential. It follows that d f p ([α]) = [ f ◦ α].
j
Now let M be a manifold and p ∈ M . Fix a chart (U, ϕ) around p. If α1 and
α2 are two curves through p then we say that α1 is equivalent to α2 if
(ϕ ◦ α1 )0 (0) = (ϕ ◦ α2 )0 (0).
We need to check that this definition does not depend on the chart chosen. Let
(V, ψ) be another chart around p and let h = ψ ◦ ϕ −1 : ϕ(U ∩ V ) → ψ(U ∩ V ) be
the change of coordinates. If α1 and α2 are equivalent with respect to ϕ, then by
applying dhϕ(p) we get
dhϕ(p) ((ϕ ◦ α1 )0 (0)) = dhϕ(p) ((ϕ ◦ α2 )0 (0))
(h ◦ ϕ ◦ α1 )0 (0) = (h ◦ ϕ ◦ α2 )0 (0)
(ψ ◦ α1 )0 (0) = (ψ ◦ α2 )0 (0)
so they are equivalent with respect to ψ as well.
6
Differential Geometry
1.2.1 Definition. Tp M , the tangent space to M at p is the set of all equivalence
classes of smooth curves α : I → M , where 0 ∈ I ⊆ R is an open interval and
α(0) = p. Two curves α1 and α2 through p are equivalent if (ϕ ◦ α1 )0 (0) =
(ϕ ◦ α2 )0 (0) for some chart (U, ϕ) containing p.
There is a linear structure on Tp M induced by the map Φϕ : Tp M → Rn : [α] →
(ϕ ◦ α)0 (0) (i.e. there is a unique linear structure on Tp M such that Φϕ is a linear
isomorphism). Again, this linear structure is defined in terms of a particular chart,
so again we need to check that it is well-defined. As above, we apply dhϕ(p) .
Tp M
DD
z
DDΦψ
z
z
DD
z
z
DD
z
}zz
!
∼
=
n
/ Rn
R
Φϕ
dhϕ(p)
Since h is a change of coordinates, dhϕ(p) is a linear isomorphism, so the linear
structures are isomorphic.
The differential d f p
Now we move to the definition of the differential of a smooth map. We take as the
definition the result noted at the end of the discussion from the beginning of the
section.
1.2.2 Definition. Let f : M → N be a smooth map between manifolds. The
differential of f at p is
d f p : Tp M → T f (p) N : [α] 7→ [ f ◦ α].
First we check that this defines a linear map. Let (U, ϕ) be a chart containing
p and (V, ψ) be a chart containing f (p). Then we have the following commutative
diagram.
Tp M
d fp
/ T f (p) N
Φϕ ∼
=
∼
= Φψ
Rn
d(ψ◦ f ◦ϕ
−1
/ Rm
)ϕ(p)
Notice that Φϕ = dϕ p with this definition of d f p , so we will no longer use the
notation Φϕ .
As in multvariable calculus, we have a Chain Rule for compositions of smooth
maps.
f
g
1.2.3 Theorem (Chain Rule). Let M −
→N−
→ P be smooth maps between manifolds, so that there are linear maps
d fp
d g f (p)
Tp M −→ T f (p) N −−−→ Tg( f (p)) P.
Tangent Space and Differentials
7
Then d(g ◦ f ) p = d g f (p) ◦ d f p .
PROOF: Exercise.
ƒ
Expressions in local coordinates
Let (U, ϕ) be a chart around p ∈ M . Let {e1 , . . . , en } be the canonical basis of Rn .
Pulling back this basis through the linear isomorphism dϕ p : Tp M → Rn gives a
“canonical” basis for Tp M (which depends on the chart chosen).
1.2.4 Definition. The canonical basis for Tp M with respect to a fixed chart (U, ϕ)
containing p is { ∂ ∂x 1 , . . . , ∂ ∂x n }, where ∂∂x i := (dϕ p )−1 (ei ).
Notation. InP
class expressions of the form “x = yi ” will be written, by which it is
n
meant “x = i=1 yi .” Namely, whenever a lonely index occurs on the righthand
side of an equality there is an implicit sum from 1 to the dimension of the space.
How does the canonical basis change when a different chart is used? Let
v ∈ Tp M and suppose v = ai ∂∂x i . Let (V, ψ) be another chart containing p and
h = ψ ◦ ϕ −1 be the change of coordinates. Suppose that h changes x i coordinates
into x i0 coordinates via
x 0j = x 0j (x 1 , . . . , x n ) = h(x 1 , . . . , x n ),
and v = ai0 ∂ ∂x 0i .
Since { ∂ ∂x 1 , . . . , ∂ ∂x n } and { ∂ ∂x 01 , . . . , ∂ ∂x 0n } are both bases of Tp M , there are (bi j )
such that
∂
∂
(dϕ p )−1 (ei ) =
= bi j
= bi j (dψ p )−1 (e j ).
i
∂x
∂ x0j
Apply dψ p to both sides and the Chain Rule to see that dhϕ(p) (ei ) = bi j e j , so
∂ x0
(bi j ) = dhϕ(p) = ( ∂ x j )i j .
i
∂ x0
∂ x0
In particular, ∂∂x i = ∂ x j ∂ ∂x 0 j , and ai0 = ∂ x i ak . Reiterating, the change of basis
i
k
matrix changing from the canonical basis induced by ϕ to the canonical basis
induced by ψ is the Jacobian at ϕ(p) of the change of coordinate map ψ ◦ ϕ −1 .
Remark. C ∞ (M , N ) denotes the space of all C ∞ maps from M to N . Suppose
f ∈ C ∞ (M , R). A element v ∈ Tp M induces a linear map v̂ : C ∞ (M , R) → R : f 7→
d f p (v). But more than that, v̂ satisfies the Leibniz rule
v̂( f g) = f (p)v̂(g) + v̂( f )g(p),
so v̂ is a so-called derivation. The tangent space can be defined to be the collection
of derivations on C ∞ (M , R). We may also write
v̂( f ) = ai
∂
∂ xi
f ◦ ϕ −1 .
8
Differential Geometry
1.3
Tangent Bundles
1.3.1 Definition. T M , the tangent bundle, is the disjoint union of all the tangent
spaces, q p∈M Tp M . Let π : T M → M be the canonical projection. π−1 (p) is the
fibre at p. If (U, ϕ) is a chart around p, define the bundle chart
ϕ T : π−1 (U) → ϕ(U) × Rn : (p, v) 7→ (ϕ(p), dϕ p (v)) = (ϕ(p), (a1 , . . . , an )).
Check that the collection of bundle charts induces a C ∞ structure on T M . The
tangent bundle is a prime example of a vector bundle.
1.3.2 Definition. Similarily, T ∗ M , the cotangent bundle, is the disjoint union of
the duals of all the tangent spaces.
Describe the charts on T ∗ M . With the proper charts, the cotangent bundle is
also a vector bundle. In classical mechanics, the tangent bundle corresponds to the
space of all positions and velocities, (q,~q), while the cotangent bundle corresponds
to the space of all positions and momentums (q, ~p). It is also know as the phase
space.
1.3.3 Definition. A vector field on M is a smooth map X : M → T M such that
π ◦ X = id.
A vector field is a prime example of a section of a vector bundle. Suppose
that (U, ϕ) is a chart around p ∈ M . Then X (p) = (ai (p) ∂∂x i ) for some smooth
functions ai : M → R. Vector fields have integral curves, which are curves γ : I →
M such that γ̇(t) = X (γ(t)). (This follows from the existence and uniqueness of
ODEs in Rn .) If M is compact then we may take I = R (i.e. there is a solution to
the ODE for all times t ∈ R). A flow is a one parameter family of diffeomorphisms
φ t : M → M (t ∈ R) such that φ t+s = φ t ◦ φs and φ t (p) = γ p (t), where γ p is
the unique solution to γ̇(t) = X (γ(t)) around p. A flow is an action of R on M by
diffeomorphisms.
1.3.4 Definition. A smooth 1-form ω on T ∗ M is a smooth map ω : M → T ∗ M
such that π ◦ ω = id. i.e. a smooth 1-form is a section of T ∗ M .
1.3.5 Proposition. Let M be a smooth manifold of dimension n, and suppose that
there are vector fields X 1 , . . . , X n such that {X 1 (p), . . . , X n (p)} is a basis of Tp M for
all p ∈ M . Then T M is “isomorphic” to M × Rn .
Here by “isomorphic” we mean diffeomorphic via a diffeomorphism with takes
fibres to fibres in a linear fashion. Manifolds M satisfying 1.3.5 are called parallelizable in the literature of differential geometry.
PROOF: For any (p, v) ∈ T M , {X 1 (p), . . . , X n (p)} is a basis of Tp M , so suppose v
has coordinates v = (ai X i (p)). Let
Φ : T M → M × Rn : (p, v) 7→ (p, (a1 , . . . , an )).
Tangent Bundles
9
Clearly Φ is a bijection and maps fibres to fibres in a linear fashion, so we need
only check that Φ and Φ−1 are smooth. Let (U, ϕ) be a chart around a point p. This
induces a chart (π−1 (U), ϕ T ) on T M around (p, v) for any v ∈ Tp M . On M × Rn
we take ϕ × id around (p, (a1 , . . . , an )). Then for any ((x i ), (b j )) ∈ ϕ(U) × Rn , let
q = φ −1 (x i ), so
∂
(ϕ × id) ◦ Φ ◦ ϕ T−1 ((x i ), (b j )) = (ϕ × id) ◦ Φ q, b j
.
∂ xj
Write X i =
P
X i j ∂∂x j , where the X i j depend on q and are smooth since X is smooth.
Write b j ∂∂x j as ai X i (q) with respect to the basis {X 1 (q), . . . , X n (q)}, we have b j =
P
P
ai X i j . Let (ci j ) be the inverse matrix of (X i j ), so that ai =
ci j b j . Then ci j
depends smoothly on q and
∂
(ϕ × id) ◦ Φ q, b j
= (ϕ × id)(q, (ci j b j )) = ((x i ), (ci j b j )).
∂ xj
ƒ
Parallelizability is special! It can be shown that every even dimensional sphere
S 2n is not parallelizable. Lie groups are always parallelizable thanks to the leftregular representation.
Lie bracket of vector fields
Let V (M ) denote the space of all smooth vector fields on M . For X , Y ∈ V (M ),
we will define the Lie bracket [X , Y ] ∈ V (M ). But first, for p ∈ M consider that
X (p) ∈ Tp M , so we may regard X as an operator on C ∞ (M , R), acting as
X ( f )(p) = XÕ
(p)( f ) = d f p (X (p)).
We say that X is a first order linear operator, and in any fixed local coordinates
(U, ϕ) X acts as
X ( f )(p) = ai (p)
∂
∂ xi
( f ◦ ϕ −1 ),
or X (p) = ai (p) ∂∂x i .
1.3.6 Definition. Let X (p) = ai (p) ∂∂x i and Y (p) = bi (p) ∂∂x i be vector fields and
define the Lie bracket of X and Y by its action on C ∞ (M , R) as
[X , Y ]( f ) = X (Y ( f )) − Y (X ( f )).
10
Differential Geometry
Notice that since f is smooth the mixed partials are equal, and
[X , Y ]( f ) = X (Y ( f )) − Y (X ( f ))
∂
∂
∂
∂
−1
−1
= aj
b
a
(
f
◦
ϕ
)
−
b
(
f
◦
ϕ
)
i
i
j
∂ xj
∂ xi
∂ xj
∂ xi
∂2
∂ bi ∂
( f ◦ ϕ −1 )
( f ◦ ϕ −1 ) + a j bi
= aj
i
∂ xj ∂ x
∂ x j∂ xi
‚
Œ
∂ ai ∂
∂2
−1
−1
− bj
(f ◦ ϕ )
( f ◦ ϕ ) + b j ai
∂ xj ∂ xi
∂ x j∂ xi
‚
Œ
∂ bi
∂ ai
∂
( f ◦ ϕ −1 ).
= aj
− bj
∂ xj
∂ xj ∂ xi
Therefore [X , Y ] is also a first order linear operator, and we are justified in writing
[X , Y ] = (a j
∂ bi
∂ xj
− bj
∂ ai ∂
)
.
∂ xj ∂ xi
1.3.7 Properties of Lie bracket. For all vector fields X , Y, Z ∈ V (M ),
(i) [·, ·] is bilinear;
(ii) [X , Y ] = −[Y, X ] (anti-commutative);
(iii) [[X , Y ], Z] + [[Y, Z], X ] + [[Z, X ], Y ] = 0 (Jacobi identity).
The vector space V (M ) with [·, ·] is an example of a Lie algebra, (i.e. a vector
space with a product satisfying (i), (ii), and (iii)).
Left invariant vector fields on Lie groups
1.3.8 Definition. Let G be a Lie group and g = Te G. For g ∈ G, the halftranslation by g is the map
L g : G → G : h 7→ hg.
Half-translations are diffeomorphisms.
1.3.9 Proposition. Any Lie group is parallelizable.
PROOF: Notice that
(d L g )e : Te G = g → Tg G,
so given ξ ∈ g we may define a vector field X ξ (g) = (d L g )e (ξ). Then X ξ : G → T G
is a smooth vector field. If {ξ1 , . . . ξn } is a basis of g (where n = dim G) then
{X ξ1 , . . . , X ξn } is a basis at every g ∈ G. By 1.3.5, G is parallelizable.
ƒ
Submanifolds
11
Now consider
(d L g )h (X ξ (h)) = (d L g )h ((d L g )e (ξ)) = d(L g ◦ Lh )e = d(L gh )e(ξ) = X ξ (gh)
for h ∈ G, since L g ◦ Lh (x) = L g (hx) = ghx = L gh (x). Therefore
(d L g )h (X ξ (h)) = X ξ (L g (h)).
1.3.10 Definition. A left-invariant vector field is a vector field satisfying. . . Let
`(G) denote the set of all left-invariant vector fields. (they all arise as X ξ for some
ξ).
1.3.11 Theorem. (`(G), [·, ·]) is a Lie algebra for any Lie group G.
PROOF: We have g → `(G) ,→ V (M ) : ξ 7→ X ξ , so we will show that `(G) is a Lie
subalgebra of V (M ).
For ξ, η ∈ g, [X ξ , X η ] ∈ `(G). Indeed, let f ∈ C ∞ (G, R).
(d L g )([X ξ , X η ])( f ) = [X ξ , X η ]( f ◦ L g )
= X ξ (X η ( f ◦ L g )) − X η (X ξ ( f ◦ L g ))
= X ξ (d L g (X η )( f )) − X η (d L g (X ξ )( f ))
= X ξ (X η ( f )) − X η (X ξ ( f ))
= [X ξ , X η ]( f )
Recalling for F : M → N , p ∈ M , v ∈ Tp M , and f ∈ C ∞ (M , R) we have
d F p (v)( f ) = d f F (p) ◦ d F p (v) = d( f ◦ F )(v) = v( f ◦ F )
1.4
ƒ
Submanifolds
1.4.1 Definition. Let M and N be two manifolds. Suppose that N ⊂ M , and let
i : N → M be the inclusion map. We say that N is an embedded submanifold if
(i) i is smooth;
(ii) d i p : Tp N → Ti(p) M is one-to-one for all p ∈ N (i.e. i is an immersion); and
(iii) i is a homoeomorphism onto its image (i.e. D ⊆ N is open in N if and only
if D = N ∩ E for some open E ⊆ M ).
1.4.2 Examples.
(i) Take M = T 2 , the 2-dimensional torus, and N = R, embedded as a line of
irrational slope. Then N is dense in M , so (iii) fails though (i) and (ii) hold.
(ii) Take M = R2 and N = R embedded in the shape of a “ρ”, but with an open
end instead of a double point (insert a diagram). Again (iii) fails but (i) and
(ii) hold.
12
Differential Geometry
We would like to decide when a set of equations define a submanifold.
1.4.3 Definition. Let f : M → N be a smooth map between two manifolds. Then
f is a submersion if d f p : TP M → T f (p) N is onto for all p ∈ M . It is an immersion if
d f p is one-to-one. A point q ∈ N is a regular value if d f p : Tp M → Tq N is onto for
every p ∈ f −1 (|[q}) (i.e. f is a submersion for every p in the preimage of q).
1.4.4 Example. Let π : Rk → R` : (a1 , . . . , ak ) 7→ (a1 , . . . , a` ) be the canonical
projection (where k ≥ `). Then π is a submersion, the canonical submersion.
1.4.5 Theorem (Inverse Function Theorem). If f : M → N be a smooth map
between manifolds then f is a local diffeomorphism at p if and only if d f p : Tp M →
T f (p) N is a linear isomorphism.
( f is a local diffeomorphism at p if there is an open neighbourhood U of p such
that f |U : U → f (U) is a diffeomorphism.)
PROOF: Use charts plus the inverse function theorem in Rn .
ƒ
1.4.6 Theorem (Preimage Theorem). Let f : M → N be a smooth map between
manifolds. If q ∈ N is a regular value, then f −1 ({q}) (when non-emtpy) is an
embedded submanifold of M of dimension dim M − dim N .
PROOF: The theorem will follow from the “local form of submersions:”
Claim. Let f : M → N , where dim M = k and dim N = `. If f is a submersion
at p then there are charts (U, ϕ) around p and (V, ψ) around f (p) such that
ψ ◦ f ◦ ϕ −1 = π, the canonical submersion.
We have the picture
M
ϕ
f
ψ
ϕ(U) ⊆ Rk
/N
h
/ ψ(V ) ⊆ R`
where h = ψ◦ f ◦ϕ −1 . Without loss of generality assume that ϕ(p) = 0, ψ( f (p)) =
0, and h(0) = 0. Then dh0 : Rk → R` is onto since f is a submersion at p. By a
linear change of coordinates we may suppose that dh0 = [I`×` |0]`×k . Define
H : ϕ(U) → Rk : (a1 , . . . , ak ) 7→ (h(a1 , . . . , ak ), a`+1 , . . . , ak ).
Then by construction dH0 : Rk → Rk is the identity map. Therefore H is a local
diffeomorphism at 0, so there is a neighbourhood W ⊆ ϕ(U) around 0 such that
Forms and Bundles
13
H|W : W → W 0 ⊆ Rk is a diffeomorphism.
MO
/N
f
ψ
ϕ −1
/ ψ(V )
:
h
tt
tt
t
H −1
tt π
tt
W 0 ⊆ Rk
WO
where π := h◦ H −1 truly is the canonical submersion. Notice that (ϕ −1 (W ), H ◦ ϕ)
is a chart around p such that
ψ ◦ f ◦ (H ◦ ϕ)−1 = ψ ◦ f ◦ ϕ −1 ◦ H −1 = h ◦ H −1 = π,
so the claim is proved.
To complete the proof, note that with the new charts (namely (ϕ −1 (W ), H ◦
ϕ) and (V, ψ), making f into a canonical submersion) x `+1 , . . . , x k is coordinate
system for f ∈ (q) around p. f −1 (q) is locally given by x 1 = · · · = x ` = 0. (Check
that this really finishs the proof.)
ƒ
1.4.7 Theorem (Whitney Embedding Theorem). Any smooth manifold of dimension n can be embedded in R2n .
2
2.1
Forms and Bundles
Differential Forms
Facts from multilinear algebra
Let V be a vector space over R of dimension n.
2.1.1 Definition. Ap (V ) is the set of alternating multilinear p-forms on V , i.e.
ω ∈ Ap (V ) if
ω : |V × ·{z
· · × V} → R,
p copies
ω is linear in each entry, and
ω(x 1 , . . . , x p ) = sg(σ)ω(x σ(1) , . . . , x σ(p) )
for all σ ∈ Sn .
There is a wedge product (or exterior product) Ap (V ) × Aq (V ) → Ap+q (V ), defined by
X
(ω ∧ η)(x 1 , . . . , x p+q ) =
sg(σ)ω(x σ1 , . . . , x σp )η(x σp+1 , . . . , x σp+q )
σ
where σ runs over all permutations of {1, 2, . . . , p + q} such that σ1 < · · · < σ p
and σ p+1 < · · · < σ p+q .
14
Differential Geometry
We have A0 (V ) = R and A1 (V ) = V ∗ (the dual vector space).
2.1.2 Properties of Wedge Product.
(i) It is bilinear and associative;
(ii) If ω1 , . . . , ωk ∈ A1 (V ) then (ω1 ∧ · · · ∧ ωk )(x 1 , . . . , x k ) = det(ωi (x j ));
(iii) If ω1 , . . . , ωn is a basis of V ∗ then {ωi1 ∧ · · · ∧ ωip | i1 < · · · < i p } is a basis of
Ap (V );
n
(iv) dim Ap (V ) = p
(v) If ω ∈ Ap (V ) and η ∈ Aq (V ), then ω ∧ η = (−1) pq η ∧ ω;
(vi) In particular, if p is odd then ω ∧ ω = 0, and specifically, the wedge product
is anti-commutative on 1-forms.
Remark. In the case p = n, so-called top dimensional forms, dim An (V ) = 1, so
the choice of a basis is really a choice of determinant (or volume form).
2.1.3 Definition. Let T : V → W be linear. There is a natural map T ∗ : Ap (W ) →
Ap (V ) defined by
(T ∗ ω)(x 1 , . . . , x p ) = ω(T x 1 , . . . , T x p ),
the pullback of T .
The pullback interacts nicely with the wedge product. Indeed,
T ∗ (ω ∧ η) = T ∗ ω ∧ T ∗ η.
Remark. For T : V → V with n = dim V and ω ∈ An (V ), we have T ∗ ω = λω
where λ = det(T ).
Back to manifolds
Let M be a manifold and (U, ϕ) be a chart giving a basis { ∂ ∂x 1 , . . . , ∂ ∂x n } of Tp M ,
for p ∈ U. The dual basis is {d x 1 , . . . , d x n }, a basis of Tp∗ M . The notation is not
accidental, d x i : Tp (M ) → R really is the differential of the coordinate functions
x i : U → R, and we have d x i ∂∂x j = δi j (check).
2.1.4 Definition. A differential p-form on M is a function x 7→ ω x , where ω x ∈
Ap (Tx M ), such that if (U, ϕ) is a chart and we write
X
ω=
f i1 ,...,ip d x i1 ∧ · · · ∧ d x ip ,
i1 <···<i p
then the functions f i1 ,...,ip are smooth. The space of all p-forms will be denoted by
Ω p (M ) (in particular Ω0 (M ) = C ∞ (M , R)).
Differential Forms
15
Contrast the above definition with the definition of a vector field. We could
have instead given q x∈M Ap (Tx M ) = Ap (M ) the structure of a vector bundle over
M and defined p-forms to be sections of this bundle.
Remark.
(i) We have a wedge product of differential forms: if ω ∈ Ω p (M ) and η ∈ Ωq (M )
then ω ∧ η ∈ Ω p+q (M ).
(ii) Smooth functions are 0-forms, so they may be wedged with p-forms. We
will not use such pompous notation as “ f ∧ ω”, and simply write f ω.
2.1.5 Definition. Let f : M → N be a smooth map. Then there is a map f ∗ :
Ω p (N ) → Ω p (M ), the pullback of f , defined by
( f ∗ ω) x (v1 , . . . , vp ) = ω f (x) (d f x (v1 ), . . . , d f x (vp )).
As in the case of the pullback of a linear map, f ∗ (ω ∧ η) = f ∗ ω ∧ f ∗ η.
Exterior differentiation
2.1.6 Theorem. There exists a unique linear operator d : Ω p (M ) → Ω p+1 (M )
(p ≥ 0) such that
(i) if f ∈ Ω0 (M ) then d f coincides with the differential of f ;
(ii) d(ω ∧ η) = dω ∧ η + (−1)deg ω ∧ dη (the Leibniz rule);
(iii) d(dω) = 0 for all ω ∈ Ω p (M ) (the chain condition).
PROOF (SKETCH): For smooth functions f we are forced to take d f =
fine
∂f
∂ xi
d x i . De-
d( f d x i1 ∧ · · · ∧ d x ip ) = d f ∧ d x i1 ∧ · · · ∧ d x ip + (−1)deg f f d(d x i1 ∧ · · · ∧ d x ip )
=
∂f
∂ xi
d x i d x i1 ∧ · · · ∧ d x ip ,
and extend d linearly to all p-forms.
Let’s see why d(dω) = 0.
d
∂f
∂ xi
d x i d x i1 ∧ · · · ∧ d x i p
=
∂2f
∂ xi∂ x j
d x j ∧ d x i d x i1 ∧ · · · ∧ d x i p .
In the double sum over i, j there a lot of cancellation since f is smooth, d x j ∧
d x i = −d x i ∧ d x j and d x i ∧ d x i = 0. In fact, nothing remains. Check that d
is unique (which is clear since we were forced to define it the way we did) and
well-defined.
ƒ
16
Differential Geometry
Consider the de Rham complex
d
d
d
d
d
Ω0 (M ) −
→ Ω1 (M ) −
→ ... −
→ Ω p (M ) −
→ Ω p+1 (M ) −
→ ...
A p-form is closed if dω = 0, and exact if there is η such that ω = dη. Notice that
exact implies closed. The de Rham cohomology is
p
H dR (M ) =
{ω ∈ Ω p (M ) | dω = 0}
{dη | η ∈ Ω p−1 (M )}
.
2.1.7 Example. Consider M = R3 . Then a 1-form is something of the form ω =
f d x + g d y + hdz, so 1-forms are in one-to-one correspondence with vector fields
F = ( f , g, h). A 2-form is something of the form
η = f d y ∧ dz + gdz ∧ d x + hd x ∧ d y,
so they are also in one-to-one correspondence with vector fields. Of course, a
3-form is just a multiple of the determinant. Now
∂f
∂f
∂g
∂g
∂g
∂f
dω = ∂ x d x + ∂ y d y + ∂ z dz + ∂ x d x + ∂ y d y + ∂ z dz + ∂∂ hx d x + ∂∂ hy d y + ∂∂ hz dz
∂g
∂f
∂g
∂f
= ∂ x − ∂ y d x ∧ d y + ∂ z − ∂∂ hx dz ∧ d x + ∂∂ hy − ∂ z d y ∧ dz
Therefore dω corresponds exactly to the curl of F . Similarly,
dη =
∂f
∂x
+
∂f
∂y
+
∂f
∂z
d x ∧ d y ∧ dz
so dη corresponds to the divergence of F . (I really wish that I had taken vector
calculus.)
2.2
Orientability and Integration
Orientable manifolds
Suppose that h : U → V is a diffeomorphism between open sets in Rn . Then we
can write coordinates for the tangent space to V as y j = h j (x 1 , . . . , x n ), where
x 1 , . . . , x n are coordinates for the tangent space of U. Then
h∗ (d y1 ∧ · · · ∧ d yn ) = det(dh x )d x 1 ∧ · · · ∧ d x n .
∂y
(Recall that det(dh x ) = det( ∂ x i ) is the Jacobian.)
j
2.2.1 Theorem (Orientability). Let M be an n-manifold. The following are equivalent.
(i) There exists a nowhere vanishing smooth differential n-form.
Orientability and Integration
17
(ii) There is a family of charts that cover M such that the determinant of the
Jacobian of coordinate changes is positive on all overlaps.
(iii) An (T M ) is isomorphic to M × R.
2.2.2 Definition. M is orientable if it satisfies any (and hence all) of the three
equivalent conditions.
PROOF: (i) if and only if (iii): Proved exactly as 1.3.5.
(i) implies (ii): Consider all charts (U, ϕ) such that d x 1 ∧ · · · ∧ d x n = f Ω,
where f > 0 and Ω is the non-vanishing n-form. We can cover M with such charts
and if (U, ϕ) and (V, ψ) are two overlapping charts then
d x 1 ∧ · · · ∧ d x n = det
∂ xi
∂ x 0j
!
d x 10 ∧ · · · ∧ d x n0
∂x
implies that det( ∂ x 0i ) > 0.
j
(ii) implies (i): We need to introduce partitions of unity.
ƒ
S
2.2.3 Theorem (Partitions of unity). For any open cover α∈A Uα = M there exists a countable collection of functions ρi ∈ C ∞ (M , R) (i = 1, 2, . . . ) such that
(i) for any i, the support {x | ρi (x) 6= 0} of ρi is compact and contained in some
Uα ;
(ii) the collection is locally finite: every point x ∈ M has a neighbourhood Wx
such that ρi 6= 0 on Wx only for finitely many i’s;
(iii) ρi ≥ 0 on M , and
P
i
ρi (x) = 1 for all x ∈ M .
The collection {ρi } is a partition of unity, and is said to be subordinate to the
cover {Uα }.
PROOF: Deferred.
ƒ
PROOF (OF ORIENTABILITY, CONTINUED):
(ii) implies (i): Cover M with compatible charts {(Uα , ϕα )} (i.e. the determinant
of the Jacobian is positive on overlaps). Take a partition of unity {ρi } subordinate
to {Uα }. Define ωi = d x 1i ∧ · · · ∧ d x ni , where the i indicates the (a?) chart that
containsP
the support of ρi . Then ρi ωi is a smooth form defined everywhere on M .
Let Ω = i ρi ωi , a smooth form on M since the sum is finite around any point. Ω
is nowhere vanishing because of the third property of partitions of unity and the
fact that the Jacobian is positive on overlaps.
ƒ
18
Differential Geometry
Integration of n-forms
2.2.4 Definition. Let M be an orientable n-manifold. Let ω ∈ Ωn (M ) have compact support (i.e. {x | ω x 6= 0} is compact).
(i) If M = U ⊆ Rn is an open set then ω = f d x 1 ∧ · · · ∧ d x n , and define
Z
ω :=
Z
U
f (x 1 , . . . , x n )d x 1 . . . d x n ,
U
where the latter is the Riemann integral.
(ii) Suppose that the support of ω is contained in U, for some chart (U, ϕ).
Then define
Z
Z
(ϕ −1 )∗ ω.
ω :=
ϕ(U)
M
(iii) Consider a partition of unity {ρi } subordinate to a cover of M by compatible
charts. Define
Z
Z
X
ω :=
ρi ω.
M
M
i
The sum is finite since ω has compact support.
Suppose that h : V → U is a diffeomorphism (where V ⊆ Rn is open). Then
h∗ ω = h∗ ( f d x 1 ∧ · · · ∧ d x n ) = ( f ◦ h) det(dh)d y1 ∧ · · · ∧ d yn .
Hence
Z
∗
h ω=
V
Z
( f ◦ h) det(dh)d y1 . . . d yn ,
V
R
and if det(dh) > 0 then the change of variables formula implies that V h∗ ω =
R
ω. In this case we say that h is orientation preserving.
U
In the second part of the definition, we need to check that if (V, ψ) is another
chart containing the support of ω then the two possible definitions of the integral
agree. Assume without loss of generality that U = V . Then if h = ψ ◦ ϕ −1 :
ϕ(U) → ψ(V ) is the change of coordinates, we have
Z
ψ(V )
(ψ−1 )∗ ω =
Z
ϕ(U)
h∗ ((ψ−1 )∗ ω) =
Z
ϕ(U)
(ψ−1 ◦ h)∗ ω =
Z
(ϕ −1 )∗ ω
ϕ(U)
In the third part one needs to check that the definition is independent of the
partition of unity.
Orientability and Integration
19
Stoke’s theorem
Let M be an oriented n-manifold and ω an n-form on M .
Notation. We will often write M n to remind us that M is an n-manifold. The
notation does not mean the Cartesian product of M with itself n times.
2.2.5 Definition. N ⊆ M n is a domain with smooth boundry (or a codimension
zero submanifold with boundry) if for all p ∈ N there is a chart (U, ϕ) around p in
M such that
n
ϕ(U ∩ N ) = ϕ(U) ∩ R−
.
(∗)
In this case we define
n
},
∂ N = {p ∈ N | ϕ satisfies (∗) and ϕ(p) ∈ ∂ R−
the boundry of N , where
n
R−
= {x ∈ Rn | x 1 ≤ 0}
n
∂ R−
= {x ∈ Rn | x 1 = 0}.
and
Remark. The boundry of N is an embedded submanifold of M . Indeed, we have
the map
n
ϕ|U∩∂ N : U ∩ ∂ N → ϕ(U) ∩ ∂ R−
,
n
and ∂ R−
= Rn−1 , so it is a chart.
The orientation of M induces an orientation on ∂ N . By flipping the sign of x 2
if necessary, we may cover M with charts that are positively oriented and satisfying
(∗). The restricted charts from the remark above give an orientation to ∂ N .
2.2.6 Definition. For p ∈ ∂ N and x ∈ Tp M , we say that v is outward directed
if dϕ p (v) has positive first coordinate. A basis {v2 . . . , vn } of Tp (∂ N ) is positively
oriented if and only if {v, . . . , vn } is a postively oriented basis of Tp M , where v is
any outward directed vector.
Check that this is a well-definition.
2.2.7 Theorem (Stoke’s Theorem).
Let N be a domain with smooth boundry in an oriented smooth n-manifold M .
Let ∂ N have the induced orientation. For every ω ∈ Ωn−1 (M ) with N ∩ supp(ω)
compact, we have
Z
Z
i∗ω =
∂N
dω,
N
where i : ∂ N → M is the inclusion map.
PROOF: Suppose first that we can prove the theorem when
(i) supp(ω) ⊆ U; where
(ii) (U, ϕ) is a positively oriented chart satisfying (∗).
20
Differential Geometry
Then cover M with positively oriented charts satisfying (∗) and take a partition of
unity {ρi } subordinate to this covering. We have
Z
Z
Z
Z
Z
Z
X
X
X
X
dω =
d
ρi ω =
d(ρi ω) =
ρi ω =
ρi ω =
ω.
N
N
i
N
i
i
∂N
∂N
i
∂N
Now let η ∈ Ωn−1
(Rn ) be the (n − 1)-form that is (ϕ −1 )∗ ω in ϕ(U) and zero
c
outside of ϕ(U). Then
Z
Z
Z
(ϕ −1 )∗ ω =
ω=
∂N
while
n
ϕ(U)∩∂ R−
Z
dω =
Z
dω =
n
ϕ(U)∩R−
N
η,
n
∂ R−
Z
dη.
n
R−
n
Therefore it suffices to prove the theorem when M = Rn , N = R−
, and ω ∈
n−1
n
Ωc (R ).
Suppose
n
X
d
ω=
f i (x)d x 1 ∧ · · · ∧ d
x i ∧ · · · ∧ dn .
i=1
Choose b > 0 such that supp( f i ) ⊆ [−b, b]n for i = 1, . . . , n. Then
i ∗ ω = ω|∂ R−n = f1 (0, x 2 , . . . , x n )d x 2 ∧ · · · ∧ d x n ,
so
Z
ω=
Z
f1 (0, x 2 , . . . , x n )d x 2 . . . d x n .
n
∂ R−
On the other hand,
n
X
∂ fi
d x1 ∧ · · · ∧ d x n,
(−1)i−1
∂
xi
i=1
dω =
so
n
X
dω =
(−1)i−1
Z
Rn
−∞
so
n
R−
i=1
When 2 ≤ i ≤ n, we have
Z∞
∂ fi
∂ xi
Z
∂ fi
∂ xi
d x1 ∧ · · · ∧ d x n.
(x 1 , . . . , x i−1 , t, x i+1 , . . . , x n )d t = 0,
Z
n
R−
∂ fi
∂ xi
d x 1 ∧ · · · ∧ d x n = 0.
Metrics
21
When i = 1,
Z
0
−∞
so
Z
n
R−
∂ f1
∂ x1
∂ f1
∂ x1
(t, x 2 , . . . , x n )d t = f1 (0, x 2 , . . . , x n ),
d x1 ∧ · · · ∧ d x n =
Z
f1 (0, x 2 , . . . , x n )d x 2 ∧ · · · ∧ d x n ,
and the theorem is proved.
2.3
ƒ
Metrics
Riemannian metrics
2.3.1 Definition. Let M be an n-manifold. A Riemannian metric on M is a function g : x ∈ M 7→ g x , where g x is a positive definite inner product on Tx M
and such that, for any chart (U, ϕ), the functions {g i j (x) = g x ( ∂∂x i , ∂∂x j ) | i, j =
1, . . . , n} are smooth. In this case the pair (M , ω) is a Riemannian manifold.
Remark.
(i) There are variations on this definition e.g. a semi-Riemannian metric requires only that the metric be non-degenerate.
(ii) We could also say that g is a smooth section of the bundle of symmetric
bilinear forms with the positivity property.
(iii) In the
case
when n = 2 and M is a surface, in classiscal notation we write
g = FE GF or as g i j d x i d x j .
A metric induces natural isomorphisms (though they depend on the metric
chosen),
Lg : T M → T ∗ M
and
L g−1 : T ∗ M → T M ,
the Legendre transform, defined by L g : (x, v) 7→ (x, w 7→ g x (v, w)).
Remark. Sometimes we write L g = [ and L g−1 = ] for these isomorphisms and
call them the musical isomorphisms. The reason for this notation is from physics,
as in abstract index notation [ lowers indices and ] raises indices. (If v = v i ∂∂x i
and p = pi d x i then pi = g i j v j , and v j = g i j pi , where g i j is the inverse of g i j .)
Notation. For v ∈ Tx M , by convention we write g x (v, v) = |v|2x = |v|2 .
Symplectic forms
2.3.2 Definition. Let M be an n-manifold. A symplectic form is a non-degenerate
smooth closed 2-form, i.e. ω ∈ A2 (T M ), dω = 0, such that ω x is a non-degenerate
bilinear form for all x ∈ M . The pair (M , ω) is a symplectic manifold.
22
Differential Geometry
For an n × n matrix A, At = −A implies that
det A = det(−A) = (−1)n det A,
so if A is invertible then n is even. Hence if (M , ω) is a symplectic manifold then
M is even dimensional.
2.3.3 Example. Let M = R2n = Rn × Rn (coordinates (qi , p j )). The canonical
Pn
symplectic form on M is ω = i=1 dqi ∧ d pi . (Check this, take the derivative, etc.)
If ω is a symplectic form on M 2n , then ω ∧ · · · ∧ ω = ωn is a top-dimensional
form. Hence if ω is non-degenerate then ωn is never zero (prove this). This
implies in particular that M is orientable. Any orientable surface is automatically
a symplectic manifold (use the form that shows orientability).
Suppose now that (M 2n , ω) is a compact symplectic manifold. Then ω is not
exact.
Indeed, if ω = dη for some smooth 1-form η, then ωn is a volume form, so
R
n
ω 6= 0 (in fact it will be greater than zero). But
M
d(η ∧ w n−1 ) = dη ∧ ωn−1 ± η ∧ d(ωn−1 ) = ωn
since ω is closed (indeed, exact), so ωn is exact. Applying Stoke’s theorem,
Z
ωn =
M
Z
d(η ∧ w n−1 ) = 0
M
2
since M has no boundry in itself. Recall that H dR
(M ) is the quotient of the closed
2-forms by the exact 2-forms, so in the case where (M , ω) is a compact symplectic
2
manifold 0 6= [ω] ∈ H dR
(M ).
k
2.3.4 Theorem (de Rham). H dR
(M ) ∼
= H k (M , R) (topological cohomology).
In the case of S 4 , H 2 (S 4 , R) = 0, so there is no symplectic form on S 4 and it
cannot be made into a symplectic manifold.
2.3.5 Example. The cotangent bundle is a very important example of a symplectic
manifold. Let N be any manifold and M = T ∗ N . A canonical 1-form λ is defined
by
λ(x,p) (ξ) = p(dπ(x.p) (ξ))
(recall that π : T ∗ N → N is the foot-point projection, so
dπ(x,p) : T(x,p) (T ∗ N ) → Tx N ).
With local coordinates (q1 , . . . , q n , p1 , . . . , pn ) on T ∗ N , we have λ = pi dq i . Locally
take ω = −dλ = dq i ∧ d pi . Then (T ∗ N , −dλ) is a (non-compact) symplectic
manifold.
Bundles
23
2.3.6 Example (Classical mechanics). Let (M , ω) be a a symplectic manifold.
For any smooth function H : M → R, dH is an exact 1-form. There exists a unique
vector field X H such that d H x (·) = ω x (X H (x), ·). This X H is called the Hamiltonian
vector field of H. If ϕ t is the flow of X H then this flow is the Hamiltonian flow of
H.
Notice that
d
dt
H(ϕ t x) = d Hϕ t x (X H (ϕ t x)) = ωϕ t x (X H (ϕ t x), X H (ϕ t x)) = 0.
Therefore “energy is preserved along the gradients of the Hamiltonian flows.”
Let (M , g) be a Riemannian manifold. For L g (x, v) = (x, p), write |p| x =de f
|v| x . Say v = v i ∂∂x i , p = pi d x i , so |p|2 = g i j pi p j and |v|2 = g i j v i v j . Then define
H(x, p) = 21 |p|2x (the Hamiltonian) and let X H be the vector field in T ∗ M , so that its
integral curves are γ : t 7→ (x(t), p(t)) ∈ T ∗ M , where γ̇(t) = X H (γ(t)). t 7→ x(t)
is a geodesic curve in M .
Define a symplectic form ω̃ = L g (−dλ) on T M . There is a canonical function
L : T M → R : (x, v) 7→ 12 |v|2x . There is an associated vector field on T M , with
integral curves γ̃ : t 7→ (x(t), v(t)), with ˙
˜γ(t) = X L (γ̃(t)). Then tλx(t) is also a
geodesic. (In fact, v(t) = ẋ(t).)
(insert diagram)
Finally, potential forces are represented by smooth functions V : M → R and
incorportated by taking H(x, p) = 12 |p|2x + V (x) (F = −∇V ).
2.4
Bundles
Vector bundles
2.4.1 Definition. Let B be a smooth manifold. A manifold E together with a
smooth submersion π : E → B is called a vector bundle of rank k over B if the
following hold.
(i) There exists a k-dimensional vector space V (called the typical fibre) such
that each fibre E p = π−1 (p) is a vector space isomorphic to V .
(ii) Any point p ∈ B has a neighbourhood U such that there exists a diffeomorphism ϕU : π−1 (U) → U × V which makes the diagram below commute.
ϕU
/ U ×V
t
t
ttt
π
t
t
t pr1
yttt
U
π−1 (U)
(iii) ϕU | Ep : E p → V is a linear isomorphism.
B is called the base space, E is called the total space, ϕU is called a trivialization,
and U is a trivializing neighbourhood.
24
Differential Geometry
2.4.2 Examples.
(i) The trivial bundle is E = B × V with π(b, r) = b.
(ii) Tangent bundles, cotangent bundles, exterior bundles Ap (T M ), . . .
V may be an R-vector space or a C-vector space.
Suppose that {Uα } is a complete family of trivializing neighbourhoods, and
ϕα : π−1 (Uα ) → Uα × V are the corresponding trivializations. Notice that
ϕβ ◦ ϕα−1 : (Uα ∩ Uβ ) × V → (Uα ∩ Uβ ) × V
has the form ϕβ ◦ ϕα−1 (b, v) = (b, ψβα (b)v), where ψβα : Uα ∩ Uβ → G L(V ). Such
ψβα is a transition function. Transition functions satisfy the cocyle conditions
(i) ψαα = idUα ;
(ii) ψαβ ψβα = idUα ∩Uβ ;
(iii) ψαβ ψβγ ψγα = idUα ∩Uβ ∩Uγ .
In fact, given a collection of functions satisfying all the properties of transition
functions, there is a unique bundle which has those functions as its transition
functions.
2.4.3 Steenrod construction. Let E = qα (Uα × Rn )/ ∼, where (x, v) ∼ ( y, w) if
and only if x = y and w = ψβα (x)v, where x ∈ Uα and y ∈ Uβ .
Very often it will happen that ψβα takes values in subgroup G < G L(V ). When
this happens we say that the bundle has structure group G.
2.4.4 Examples.
(i) For E = B × Rn , the transition function is ψα (x) = I n .
(ii) M is orientable if and only if the tangent bundle T M has structure group in
G L+ (k, R), linear maps with positive determinant.
(iii) The structure group of the Möbius band over S 1 is {±1} ∼
= Z2 .
(iv) O(k), SO(k), . . .
Principal bundles
2.4.5 Definition. Let G be a Lie group and P be a smooth manifold. A smooth
action (in this case a smooth right action) is just an action P × G → P : (p, g) 7→ pg
which is also a smooth map. The action is a free action if pg = p implies g = 1.
2.4.6 Definition. Let B be a smooth manifold. A principal bundle (or principal
G-bundle) is a manifold P together with a smooth submersion π : P → B together
with a smooth right free action satisfying the following conditions.
(i) π(p g) = π(p) for all p ∈ P and g ∈ G (i.e. the fibres of π are the orbits of
G);
Bundles
25
(ii) for any b ∈ B there is a neighbourhood U of b and a diffeomorphism ϕU :
π−1 (U) → U × G such that pr1 ◦ ϕU = π|π−1 (U) ;
(iii) the actions are “intertwined” by ϕU , i.e. for all h ∈ G, ϕU (ph) = (b, gh),
where (b, g) = ϕU (p) and π(p) = b ∈ U.
As in the case of vector bundles, we have trivializing neighbourhoods and
transition functions. Let {Uα } be a complete family of trivializing neighbourhoods,
and consider the transition function defined by
ϕβ ◦ ϕα−1 (b, g) = (b, ψβα (b, g)).
As before, for each b ∈ Uα ∩ Uβ , ϕβα (b, ·) is a map from G to G. Since the action
is intertwined,
ψβα (b, g)h = ψβα (b, gh).
Taking g = 1, we have ψβα (b, 1)h = ϕβα (b, h), so ψβ,α (b, ·) (as a map from G to
G) is left-translation on G by the element ψβα (b, 1). As is usual in geometry, we
now simplify the notation by renaming
ϕβα : Uα ∩ Uβ → G : b 7→ ψβα (b, 1),
the transition function associated with the principal bundle. Transition functions
for principal bundles satisfy the cocycle conditions (namely, ψαβ ◦ψβγ ◦ψβγ = id).
As in the case of vector bundles, if we know the transition functions then we can
recover the bundle by the Steenrod construction. Take P = qα (Uα × G)/ ∼, where
(b, h) ∼ (b0 , h0 ) if and only if b = b0 and h0 = ψβα (b)h.
Suppose that G ⊆ G L(k) (or, what amounts to the same thing, a representation
of G in G L(k)) and we have a collection {ψβα : Uα ∩ Uβ → G} of transistion
functions which satisfy the cocyle conditions. Then there is an associated vector
bundle E and an associated principal bundle P. In this case E and P are said to be
associated bundles
2.4.7 Definition. A section of a vector bundle E (resp. principal bundle P) is a
smooth map s : B → E (resp. P) such that π ◦ s = id.
2.4.8 Example (Hopf bundle). Let B = CP1 = (C2 \ {(0, 0)})/C∗ , the collection
of (complex) lines through the origin in C2 . We have charts CP1 = U1 ∪ U2 , where
z
Ui = {[z1 : z2 ] ∈ CP1 | zi 6= 0}, with coordinates z : U1 → C : [z1 : z2 ] 7→ z2 and
η : U2 → C : [z1 : z2 ] 7→
z1
.
z2
1
On U1 ∩ U2 we have η = 1z .
Let E be the disjoint union of the (complex) lines through the origin in C2 , a
bundle over CP1 with trivial submersion π : E → CP1 : ` 7→ `. E is the tautological
bundle of CP1 . A point in E will be written as (ωz1 , ωz2 ), with |z1 |2 + |z2 |2 6= 0.
Trivializations are
p
ϕ1 : π−1 (U1 ) → U1 × C : (ω, ωz) 7→ ([1 : z], ω 1 + |z|2 )
and
ϕ2 : π−1 (U2 ) → U2 × C : (ωη, ω) 7→ ([η : 1], ω
p
1 + |η|2 ).
26
Differential Geometry
It can be shown that, for ([1 : z], ω) ∈ (U1 ∩ U2 ) × C,

ϕ2 ◦ ϕ1−1 ([1
: z], ω) = ϕ2  p

ω
1 + |z|2
,p
ωz
1 + |z|2



|η|ω

, p
= ϕ2  p
1 + |η|2 η 1 + |η|2
|η|
z
= [η : 1],
ω = [1 : z], ω
η
|z|
Therefore ψ21 ([1 : z]) =
z
|z|
|η|ω
∈ U(1) ⊆ C∗ . This transition function gives a vector
bundle or a principal bundle over CP1 , both of which are called the Hopf bundle.
2.4.9 Example. S 3 embeds in C2 as S 3 = {(z1 , z2 ) ∈ C2 | |z1 |2 + |z2 |2 = 1}. For
e iθ ∈ U(1),
S 3 × U(1) → S 3 : ((z1 , z2 ), e iθ ) 7→ (e iθ z1 , e iθ z2 )
π
is often called the Hopf map. The orbits are circles in S 3 , and we have S 3 −
→ CP1 =
2
S . This is an example of a non-trivial principal bundle. (I don’t understand. . . )
Pullback bundles, morphisms, and automorphisms
For this section we will phrase everything in terms of vector bundles, but the
definitions work, with appropriate modification, for principal bundles as well.
π
2.4.10 Definition. Let E −
→ B be a vector bundle and f : M → B be a smooth
map. Let f ∗ E = {(x, e) ∈ M × E | π(e) = f (x)}, F (x, e) = e, and π0 (x, e) = x.
Then the following diagram commutes.
f ∗E
F
π0
M
f
/E
/B
π
f ∗ E is called the pullback bundle of E through f , a vector bundle over M .
From the point of view of transition functions, if
ψβα : Uα ∩ Uβ → G ⊆ G L(V )
π
then
π0
f ∗ ψβα = ψβα ◦ f .
2.4.11 Definition. Suppose that E −
→ B and E 0 −
→ B 0 are two vector bundles over
the same vector space V and f : B → B 0 is smooth. A smooth map F : E → E 0
if a bundle morphism covering f if for any b ∈ B, F restricts to a linear map
Connections
27
F | E b : E b → E 0f (b) and such that the following diagram commutes.
E
π
B
F
/ E0
π0
f
/ B0
If F is a diffeomorphism and F | E b : E b → E 0f (b) is a linear isomorphism for all b ∈ B
then we say that F is a bundle isomorphism. An isomorphism E → E covering the
identity map is a bundle automorphism. The groups of all autmorphisms is denoted
Aut(E).
From a local point of view, take trivializing neighbourhoods U and U 0 such
that f (U) ⊆ U 0 , ϕU : π−1 (U) → U × V , and ϕU0 0 : (π0 )−1 (U 0 ) → U 0 × V . Let
FU = ϕU0 0 ◦ F ◦ ϕU−1 , so that FU (b, v) = ( f (b), h(b)v), where h : U → L (V, V ).
2.4.12 Example. Take E = B × V , the trivial bundle. Then one chart suffices, so
the local behavior determines the map, and we have Aut(E) = C ∞ (B, G L(V )).
Remark.
(i) If the bundle has structure group G ⊆ G L(V ) then we often only care about
AutG (E) ⊆ Aut(E), where AutG (E) are those automorphisms for which when
written in trivializations giving the G-struture, h above takes values in G.
(ii) Using bundle automorphisms one can change transition functions.
ψ0βα = hα ψβα h−1
α
and
ϕα0 (e) = hα (π(e))ϕα (e).
This change of trivializations is analogous to the change of basis in G L(V ).
2.5
Connections
First definition
We would like to generlize the fact that the derivative of a vector field (vi (x)) in
π
Rm is another vector field in Rm , namely (vi0 (x)). If E −
→ B is a vector bundle and
s : B → E is a section then we would like a way of differentiating s in such a way
that the derivative is also a section.
With the notation above, suppose dim B = n and V = Rm . Let U ⊆ B be
a trivializing neighbourhood (so there is ϕU : π−1 (U) → U × Rm ), and let x k ,
k = 1, . . . , n be local coordinates in U. Let a j , j = 1, . . . , m be canonical coordinates
in Rm . Using the trivialization, Tp E (for p ∈ π−1 (U)) has a basis { ∂ ∂x k , ∂∂a j }. For
π : E → B, dπ p : Tp E → Tπ(p) B is a submersion and we call its kernel the vertical
subspace. The vertical subspace is spanned by { ∂∂a j }.
2.5.1 Definition. A subspace S p ⊆ Tp E is called a horizontal subspace if
S p ∩ ker(dπ p ) = {0}
and
S p ⊕ ker(dπ p ) = Tp E.
28
Differential Geometry
We will not simply take S p = { ∂∂x j }, since this depends heavily on the choice of
local coordinates; notice that the vertical subspace does not depend on the local
coordinates.
2.5.2 Lemma. If θ 1 , . . . , θ m ∈ (Rm+n )∗ are linear functionals then
dim
\
m
ker θ i
=n
i=1
if and only if {θ 1 , . . . , θ m } is linearly independent.
PROOF: Exercise (or possibly omitted).
ƒ
Let θ p1 , . . . , θ pm be m linearly independent covectors (i.e. elements of (Tp E)∗ ).
Tm
Take S p = i=1 ker(θ pi ). (Tp E)∗ has basis {d x k , da j }, so we can write θ pi = f ki d x k +
g ij d a j . If S p is a horizontal subspace then θ pi (v) = 0 for all i implies v = 0, for
all vertical vectors v. Restating, if S p is a horizontal subspace and v = c j ∂∂a j then
0 = θ pi (v) = g ij c j for all i implies c j = 0. It follows that S p is horizontal if and
only if (g ij )i j is an invertible m × m matrix. Say (g ij )−1 = (d ji ). Replace θ i by θ̃ i by
multiplying by (d ji ), i.e. θ i = d ji θ j = da i + eki d x k . This change does not alter S p .
Now let p vary, so the eki become functions of p. If eki (p) are smooth functions of
p then we say that S p varies smoothly with p.
The following proposition summarizes the above discussion.
2.5.3 Proposition. Let S = S p (p ∈ E) be any smooth field of horizontal subspaces
in T E. Let x k , a j be localTcoordinates on π−1 (U) arising from a local trivialization
m
j
(as above). Then S p = i=1 ker(θ pi ), where θ pi = da j + ek (x, a)d x k for smooth
j
functions ek (x, a) which are uniquely determined by the trivialization.
2.5.4 Definition. A smooth field of horizontal subspaces S p is a connection if in
every local trivialization, the functions eki (x, a) are linear in the fibre variables,
i.e. eki (x, a) = Γijk (x)a j , where Γijk : U ⊆ B → R are smooth functions called the
coefficients of the connection S p .
Notation. We write θ i = da i + Aij a j , where Aij = Γijk d x k . (Aij )i j is an m × m matrix
of 1-forms on U.
Coming up, an “index festival”. . .
Transformation law for connections and the second definition
Let U 0 be another trivializing neighbourhood with trivialization ϕ 0 : π−1 (U 0 ) →
Rm , and x k0 , a i0 coordinates in U 0 × Rm .
Notation. A prime “0 ” refers to U 0 e.g. the transition matrix on U ∩ U 0 is ψi0i and
its inverse is ψii0 .
Connections
29
Let x k0 = x k0 (x). Then a i0 = ψi0i a i , so
d a i0 = dψi0i a i + ψi0i da i =
and d x k0 =
∂ x k0
d x k.
∂ xk
θ = da
i0
d xk
a i + ψi0i da i ,
Therefore

i0
∂ ψi0i
+ Γi0j0k0 a j0 x k0
so
=

ψii0 θ i0 =
ψi0i da i
+
∂ ψi0j
+ Γi0j0k0
d xk
∂ x k0
∂

j0
ψ  a j d x k,
xk j

∂ ψi0j
k0
j0 ∂ x  j
i i0
a d x k.
+
ψ
Γ
ψ
d a i + ψii0
j
i0
j0k0
d xk
∂ xk
From this is follows that
Γijk
=
j0 ∂
Γi0j0k0 ψii0 ψ j
∂
so
Aij = Γijk d x k
x k0
+ ψii0
xk
∂ ψi0j
d xk
j
Ai0j0 = ψi0i Aij ψ j0 + ψi0j dψij0 .
and
(1)
(2)
Rewritting, if Aϕ is the matrix of 1-forms in U and Aϕ0 is the matrix of 1-forms in
U 0 , then
Aϕ0 = ψAϕ ψ−1 + ψ(dψ−1 ) = ψAϕ ψ−1 − (dψ)ψ−1 .
(3)
2.5.5 Theorem. Any system of functions Γijk (1 ≤ k ≤ n, 1 ≤ i, j ≤ m) attached
to local trivializations satisfying the transformation law (1) defines a connection
A on E whose coefficients are Γijk .
Third definition
π
Notation. For any vector bundle E −
→ B, Γ(E) denotes the space of sections s :
B → E (so s(x) ∈ E x ).
2.5.6 Definition. A connection is an R-linear map ∇ : Γ(E) → Γ(T ∗ B ⊗ E) such
that
∇( f s) = d f ⊗ s + f ∇s
Leibniz rule
for f ∈ C ∞ (B, R) and s ∈ Γ(E).
Or better,
2.5.7 Definition. A connection is a map ∇ : Γ(T B) × Γ(E) → Γ(E) which we write
as ∇(X , s) = ∇X s and such that
(i) ∇ f X +gY s = f ∇X s + g∇Y s for all f , g ∈ C ∞ (B, R) and X , Y ∈ Γ(T B);
(ii) ∇X (s1 + s2 ) = ∇X s1 + ∇X s2 and ∇X (cs) = c∇X s for all si ∈ Γ(E) and c ∈ R;
30
Differential Geometry
(iii) ∇X ( f s) = X ( f )s + f ∇X s (= d f (X )s + f ∇X s).
Recall that if V and W are vector spaces then there is a canonical linear isomorphism
V ∗ ⊗ W → Hom(V, W ) : f ⊗ w 7→ “v 7→ f (v)w”.
Whence T ∗ B ⊗ E ∼
= Hom(T B, E), so a section s : B → T ∗ B ⊗ E gives linear maps
s x : Tx B → E x .
We can think of Γ(T ∗ B ⊗ E) as the E-valued 1-forms, and we will denote it
1
ΩB (E) by analogy with the notation for forms. (Our old notation becomes Ω1 (B) =
Ω1B (B × R).) Sections of E are thought of as E-valued 0-forms, denoted Ω0B (E)
2.5.8 Definition. Let ΩBr (E) be the collection of E-valued r-forms, alternating
multilinear maps
T B × · · · × Tx B → E x .
|x
{z
}
r
With these definitions, ∇ : Ω0B (E) → Ω1B (E). This will of course be generalized
later.
The definitions 2.5.6 and 2.5.7 at the beginning of this section are see to be
the same via the identification ∇X (s) = (∇s)(X ).
Bringing it all together
(Insert several puns on the word “connection” here.)
Let A be a connection (i.e. Aij = Γijk d x k with the correct transformation law).
Work in a local trivialization, and define ∇ := dA by dAs = ds + As. Locally we may
write s(x) = (s1 (x), . . . , s m (x)) (a vector in Rm ), so
Œ
Œ
‚‚ 1
m
∂s
∂s
k
k
1 j
m j
1
m
+ Γ jk s d x , . . . ,
+ Γ jk s d x .
dA(s , . . . , s ) =
∂ xk
∂ xk
Now dA is well-defined since if we take another trivialization U 0 with transition
matrix ψ from U 0 to U, then from above we have A = ψA0 ψ−1 − (dψ)ψ−1 . We
also have s = ψs0 . Checking,
dAs = ds + As = d(ψs0 ) + A(ψs0 ) = (dψ)s0 + ψds0 + (ψA0 ψ−1 − (dψ)ψ−1 )(ψs0 )
= (dψ)s0 + ψds0 + ψA0 s0 − (dψ)s0 = ψds0 + ψA0 s0 = ψ(dA0 s0 )
Remark. From the definition of ∇, it follows that ∇ is a local operator, i.e. if
s ∈ Γ(E) vanishes on an open set U then ∇s vanishes on U as well. Indeed, if
x ∈ U then let V be a neighbourhood of x such that V ⊆ U. Let α be a C ∞ cutoff function such that 0 ≤ α ≤ 1, α|V ≡ 1, and supp(α) ⊆ U. Then αs = 0, so
0 = ∇(αs) = dα ⊗ s + α∇s by linearity and the Leibniz rule. Evaluating at x shows
(∇s)(x) = 0.
This implies that a connection ∇ on E also defines a connection on E|U , where
U is any open set of B.
Connections
31
2.5.9 Theorem. Any connection ∇ arises as dA for some A.
PROOF: Take a trivialization over U ⊆ B. Local sections
Γ(E|U ) = {s : U → E | s(x) ∈ E x }
are just smooth functions U → Rm . We have a frame of sections {e j }, the constant
sections which map everything to (0, . . . , 0, 1, 0, . . . , 0) ∈ Rm . Then we can write
an arbitrary section as s = s j e j . Therefore
∇s = ∇(s j e j ) = ds j ⊗ e j + s j ∇e j .
But for some Γ, we can write ∇e j = (Γijk d x k )ei . Let Aij = Γijk d x k . We now check
that ∇s = dAs.
∇s = ds j ⊗ e j + s j Γijk d x k ei = (ds i + s j Γijk d x k )ei = dAs
To finish the proof one has to check that the Γijk (in A) transform in the correct
way.
ƒ
2.5.10 Example. Let M n ⊆ Rn+k , and let s ∈ Γ(T M ) (so s is a vector field).
Then s : M → T M such that s(x) ∈ Tx M ⊆ Tx Rn+k = Rn+k . For v ∈ Tx M , it
makes sense to write ds x , and ds x (v) ∈ Rn+k . But this may have jumped out
of Tx M , so orthogonal projection onto Tx M brings it back. Define (∇ v s)(x) to
be the orthogonal projection of d x s(v) onto Tx (v). As the notations suggests,
∇ : Γ(T M ) × Γ(T M ) → Γ(T M ) is a connection on T M . This is the Levi-Civita
connection of the metric induced on M by Rn+k .
π
Let E −
→ B and α : I → B be a smooth curve, and s : I → E be a section along
α, i.e. s(α(t)) ∈ Eα(t) for all t ∈ I. One can introduce an operator dDt which takes
s into another section along α and such that
(i)
D
(s
dt 1
(ii)
D
(f
dt
+ s2 ) =
Ds1
dt
s) = f˙s + f
+
Ds
,
dt
Ds2
;
dt
where f is a smooth function of t; and
(iii) If s(t) = ω(α(t)), where ω ∈ Γ(E), then
Ds
dt
= ∇α̇ ω.
Let α : I → U ⊆ B, where U is a trivializing neighbourhood, with trivialization
ϕU : π−1 (U) → U × Rm , with s(t) = (s1 (t), . . . , s m (t)) (so s = s j e j ). We must have
D
dt
(s j e j ) = ṡ j e j + s j ∇α̇ e j
De
and if α = (α1 , . . . , αn ) then d tj = ∇α̇ e j = Γijk d x k (α̇)ei = Γijk α̇k ei . Bringing it all
together,
D j
(s e j ) = (ṡ i + s j α̇k Γijk )ei .
dt
It can be shown that this is the unique local definition of
D
.
dt
32
Differential Geometry
2.5.11 Definition.
(i) s ∈ Γ(E) is parallel (or covariant-constant) if ∇s = 0.
(ii) If s is a section along a curve α : I → B is parallel if
Ds
dt
= 0.
Notice that dDst = 0 means that ṡ i + s j α̇k Γijk = 0 for every i. This is a linear
system of ODE’s. Take s0 ∈ Eα(t 0 ) , so there is a unique parallel section s along
α such that s(t 0 ) = s0 . This defines P : Eα(t 0 ) → Eα(t 1 ) , P(s0 ) = s(t 1 ), a linear
isomorphism. Such a thing is a parallel transform. The velocity vectors ṡ of the
parallel sections t 7→ s(t) ∈ E are live in the horizontal subspaces at s(t) (exercise,
use the forms θ i = d a i + Γijk a j d x k to show that θ i (ṡ) = 0 if s is parallel).
π
Remark. If ∇1 and ∇2 are connections on E −
→ B, then ∇1 ( f s) − ∇2 ( f s) =
1
2
1
2
∞
f (∇ s − ∇ s), so s 7→ ∇ s − ∇ s is a C -linear map from sections to E-valued
1-forms. We may think of having ∇1 − ∇2 ∈ Ω1B (End E). Whence the space of
connection is an affine space.
In a similar way we have ΩBr (End E), the End E-valued r-forms. In priniple,
they act on E-valued sections (or `-forms), getting ΩBr (End E) × Ω`B (E) → ΩBr+` (E).
2.6
Curvature
“I’m not going to tell you what it is, I’m going to tell you where it lives. . . ”
Recall the covariant derivative dA : Ω0B (E) → Ω1B (E). We would like to extend
it dA : ΩBr (E) → ΩBr+1 (E) analogously to how we extended d. For σ ∈ ΩBr (E) and
ω ∈ Ωq (B) (and ordinary q-form), we require
dA(σ ∧ ω) = dAσ ∧ ω + (−1) r σ ∧ dω.
Locally in a trivialization we get dAσ = dσ + A ∧ σ. We have
dA
dA
dA
dA
dA
Ω0B (E) −
→ Ω1B (E) −
→ ... −
→ ΩiB (E) −
→ ... −
→0
But what is dA2 ?. For σ ∈ ΩBr (E), we have
dA(dAσ) = dA(dσ + A ∧ σ)
= d(dσ + A ∧ σ) + A ∧ (dσ + A ∧ σ)
= 0 + dA ∧ σ + (−1)A ∧ dσ + A ∧ dσ + A ∧ A ∧ σ
= (dA + A ∧ A) ∧ σ
Let F = dA + A ∧ A, so that dA(dAσ) = F ∧ σ. Notice that dA dA( f σ) = f dA dAσ, so
F ∈ Ω2B (End E). F is the curvature, the deviation of our chain from being a chain
complex.
2.6.1 Definition. A flat connection is a connection with F = 0.
Riemannian Metrics
33
2.6.2 Example. Let E = B × R (or B × Rm if you wish). Then ΩBr (E) = Ω r (B)
we may take dA to simply be d, i.e. dAω = dω, the trivial connection or product
connection. Since d 2 = 0, the curvature is zero and the connection is flat.
Suppose that Aij = Γijk d x k , and A = Ak d x k , where Ak is an m × m matrix of
functions. Then
F = dA + A ∧ A
= d(Ak d x k ) + (Ai d x i ) ∧ (Ak d x k )
=
∂ Ak
d x i ∧ d x k + Ai Ak d x i ∧ d x k
∂ xi
1 ∂ Ak
∂ Ai
=
−
+ [Ai , Ak ] d x i ∧ d x k
2 ∂ xi
∂ xk
(C1)
Finally, dA also extends naturally to ΩBr (End E), but how? Let µ ∈ ΩBr (End E),
σ ∈ Ω`B (E), so that µ ∧ σ ∈ ΩBr+` (E). We would like to define dA so that
dA(µ ∧ σ) = (dAµ) ∧ σ + (−1) r µ ∧ dAσ.
But we may take that as the definition. Locally we have
(dAµ) ∧ σ = dA(µ ∧ σ) − (−1) r µ ∧ dAσ
= d(µ ∧ σ) + A ∧ µ ∧ σ − (−1) r µ ∧ (dσ + A ∧ σ)
= (dµ) ∧ σ) + (−1) r µ ∧ dσ + A ∧ µ ∧ σ
− (−1) r µ ∧ dσ − (−1) r µ ∧ A ∧ σ)
= (dµ + A ∧ µ − (−1) r µ ∧ A) ∧ σ
Therefore dAµ = dµ + A ∧ µ − (−1) r µ ∧ A.
2.6.3 Theorem (Bianchi Identity). dA F = 0
PROOF: Since dA dAσ = F ∧ σ,
(dA F ) ∧ σ = dA(F ∧ σ) − F ∧ dAσ = dA(dA dAσ) − dA dA(dAσ) = 0
3
3.1
ƒ
Riemannian Metrics
Metric Connections
π
Let E −
→ B, A be a connection, and 〈·, ·〉 a Riemannian metric on E over R (or a
Hermitian metric on E over C).
3.1.1 Definition. A metric connection (or orthogonal connection or unitary connection) is a connection such that for any X ∈ V (B) and s1 , s2 ∈ Γ(E),
X 〈s1 , s2 〉 = 〈∇X s1 , s2 〉 + 〈s1 , ∇X s2 〉.
34
Differential Geometry
Remark. If E is endowed with a Riemannian metric then we may choose trivializations ϕ : π−1 (U) → U × Rm such that the vectors ei (p) such that they are mapped
by ϕ to the canonical basis in Rm are orthogonal (exercise: check this (use GramSchmidt method)). Such a trivialization is called an orthogonal trivialization.
3.1.2 Proposition. A metric connection has skew-symmetric matrix of coefficients
in any orthogonal trivialization.
PROOF: Take ei , an orthonormal set of sections in the trivialization. Then for any
vector field X ,
0 = X 〈ei , e j 〉
= 〈∇X ei , e j 〉 + 〈ei , ∇X e j 〉
= 〈Aki (X )ei , e j 〉 + 〈ei , Akj (X )e j 〉
j
= Ai (X ) + Aij (X ).
ƒ
3.1.3 Corollary. Let A be a metric curvature. Then in an orthogonal local trivialization, the matrix of F is also skew-symmetric.
(In the form (C1) for curvature one can always swap i and k and get the
negative of what one started with, but the corollary refers to taking the transpose,
so we see a different kind of skew-symmetry.)
3.2
Levi-Civita connection
Let M be a manifold with a Riemannian metric g (also written 〈·, ·〉). A connection
on M is a connection on T M , ∇ : Γ(T M ) → Γ(T ∗ ⊗ T M ). We have Γ(T M ) =
V (M ), the vector fields on M . Notice that we may consider 〈X , Y 〉 a function on
M , taking value, for p ∈ M , 〈X (p), Y (p)〉 p .
Recall the compatibility conditions for a connection to be a metric connection.
They imply (?) that
Z〈X , Y 〉 = 〈∇ Z X , Y 〉 + 〈X , ∇ Z Y 〉.
Notice that
∇X ( f Y ) − ∇ f Y (X ) = X ( f )Y + f (∇X Y − ∇Y X ),
while
[X , f Y ] = X ( f )Y + f [X , Y ],
so T (X , Y ) := ∇X Y − ∇Y X − [X , Y ] is a tensor, called the torsion of ∇. Locally,
if T ( ∂∂x i , ∂∂x j ) = Tikj ∂ ∂x k then we may write T (X , Y ) = X i Y j Tikj ∂ ∂x k by C ∞ (M , R)linearity. If the torsion is zero then the connection is said to be symmetric.
3.2.1 Theorem. For any Riemannian manifold, there is a unique connection ∇
such that
Curvature revisited
35
(i) ∇ is compatible with g; and
(ii) ∇ is symmetric, i.e. ∇X Y − ∇Y X = [X , Y ].
This connection is the Levi-Civita connection of M .
PROOF: Suppose that ∇ exists. Then
(i) X 〈Y, Z〉 = 〈∇X Y, Z〉 + 〈Y, ∇X Z〉;
(ii) Y 〈X , Z〉 = 〈∇Y X , Z〉 + 〈X , ∇Y Z〉;
(iii) Z〈X , Y 〉 = 〈∇ Z X , Y 〉 + 〈X , ∇ Z Y 〉.
Then (i) + (ii) - (iii) gives
X 〈Y, Z〉 + Y 〈X , Z〉 − Z〈X , Y 〉
= 〈[X , Z], Y 〉 + 〈[Y, Z], X 〉 + 〈[X , Y ], Z〉 + 2〈X , ∇Y X 〉
(4)
If ∇0 is another connection then 〈Z, ∇Y X 〉 = 〈Z, ∇0Y X 〉 for all Z, so ∇Y X = ∇0Y X
for all X , Y . To show existence, define ∇ by (4) and check that ∇ is compatible
and symmetric.
ƒ
In local coordinates, if ∇
Γkij g`k
3.3
∂
∂ xj
∂
∂ xi
=
1
= Γkij ∂ ∂x k , then
‚
∂ g jk
∂ xi
2
+
∂ g ki
∂ xj
−
∂ gi j
∂ xk
Œ
.
Curvature revisited
From now on we deal with a Levi-Civita connection on a Riemannian manifold.
Denote the curvature of ∇ by R ∈ Ω2M (End T M ). For X , Y, Z ∈ V (M ), R(X , Y )Z ∈
V (M ). This is C ∞ (M , R)-linear in all three variables. Further, R(X , Y ) = −R(Y, X ).
Remark. If M = Rn and g is the usual inner product then Γkij ≡ 0, whence R ≡ 0.
(So Rn is flat, surprise, surprise. . . )
Locally, write R( ∂ ∂x k , ∂∂x l ) ∂∂x j = Rijkl ∂∂x i . We have seen that Rijkl = −Rijlk . We
have the expression
∂ Ak
1 ∂ Al
R=
−
+
[A
,
A
]
d xk ∧ d xl,
k
l
2 ∂ xk
∂ xl
where A = Ak d x k , so
∂
R
∂
xk
,
∂
∂
xl
=
∂ Al
∂
xk
−
∂ Ak
∂ xl
+ [Ak , Al ].
There is a direct formula for Rijkl in terms of the g i j and their first and second
derivatives.
36
Differential Geometry
3.3.1 Proposition. R(X , Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X ,Y ] Z.
The conclusion of this proposition may be taken as the definition of the curvature, given a Levi-Civita connection.
PROOF: It suffices to check this equation locally. Check first for { ∂∂x i }, and then
use C ∞ (M , R)-linearity. Use the fact that ∇
∂
∂ xi
=
∂
∂ xi
+ Ai (exercise).
ƒ
Warning: some authors/books will use the convention that
R(X , Y )Z = ∇Y ∇X Z − ∇X ∇Y Z + ∇[X ,Y ] Z
i.e. with the opposite sign.
Recall that R(X , Y ) = −R(Y, X ), or Rijkl = Rijlk . Many times we will express
the symmetry conditions in terms of the 4-tensor 〈R(X , Y )Z, T 〉, locally given by
q
R i jkl = g iq R jkl .
3.3.2 Proposition (First Bianchi Identity).
R(X , Y )Z + R(Y, Z)X + R(Z, X )Y = 0
PROOF: Since the left hand side is linear, it suffices to check the equation locally
ƒ
on a basis { ∂∂x i } with [ ∂∂x i , ∂∂x j ] = 0.
Locally this identity is Rijkl + Ril jk + Rikl j = 0.
3.3.3 Proposition.
(i) 〈R(X , Y )Z, T 〉 = −〈R(Y, X )Z, T 〉 = −〈R(X , Y )T, Z〉
(ii) 〈R(X , Y )Z, T 〉 = −〈R(Z, T )X , Y 〉
Locally, R i jkl = −R i jlk = −R jikl and R i jkl = R kli j .
PROOF:
(i) The first equality we have done before. The second equality is a consequence
of 3.1.3 since the Levi-Civita connection is a metric connection, so the map
Z 7→ R(X , Y )Z is anti-symmetric.
(ii) From the first Bianchi identity,
〈R(X , Y )Z, T 〉 + 〈R(Y, Z)X , T 〉 + 〈R(Z, X )Y, T 〉 = 0,
〈R(Y, Z)T, X 〉 + 〈R(Z, T )Y, X 〉 + 〈R(T, Y )Z, X 〉 = 0,
〈R(Z, T )Y, Y 〉 + 〈R(T, X )Z, Y 〉 + 〈R(X , Z)T, Y 〉 = 0,
〈R(T, X )Z, Z〉 + 〈R(X , Y )T, Z〉 + 〈R(Y, T )X , Z〉 = 0.
Add and use part (i).
ƒ
Sectional, Ricci, and Scalar curvature
3.4
37
Sectional, Ricci, and Scalar curvature
Sectional Curvature
p
For u, v ∈ Tx M (linearly independent), let |u ∧ v| = |u|2 |v|2 − 〈u, v〉2 , and let
σ ⊆ Tx M be the 2-dimensional subspace spanned by u and v. Define
K x (σ) :=
〈R(u, v)v, u〉
|u ∧ v|2
,
the sectional curvature. Check that K x (σ) depends only on σ and not the basis
chosen. In fact K x (σ) determines R, i.e. if R0 is another multilinear map with the
same symmetry properties as R and K = K 0 then R = R0 .
Remark. If dim M = 2 then there is only one plane to chose, spanned by { ∂ ∂x 1 , ∂ ∂x 2 },
so we get
〈R( ∂ ∂x 1 , ∂ ∂x 2 ) ∂ ∂x 2 , ∂ ∂x 1 〉
R1212
=
Kx =
EG − F 2
EG − F 2
where E = | ∂ ∂x 1 |2 , G = | ∂ ∂x 2 |2 , and F = 〈 ∂ ∂x 1 , ∂ ∂x 2 〉. But there is a formula for R1212
in terms of Γijk , so we get a formula for K in terms of the Christofel (sp?) symbols.
It is the same expression for K that arises in the Teoreme Egregium of Gauss.
Ricci Curvature
For u, v ∈ Tx M , define
Ric g (u, v) := Tr(w 7→ R(w, u)v),
q
the Ricci curvature. It is a symmetric bilinear form and Rici j = R iq j = g lq R il jq
(symmetry follows from the symmetry of R).
Scalar Curvature
The scalar curvature at a point x is the trace of the symmetric linear form associated with Ric g , i.e. write Ric g (u, v) = 〈Q x (u), v〉, where Q x (u) is a symmetric map,
and define s g (x) = Tr(Q x ). Then s g (x) = g ik Ricik .
An Einstein manifold is a Riemannian manifold (M , g) for which Ric g = λg for
some λ ∈ R. An Einstein manifold is Ricci flat if λ = 0. A Ricci flow is a flow of the
∂g
form ∂ tt = −2 Ric g t ( − λg t , here we think of the manifold being fixed, and the
Riemannian metric varying with time, and λ is a “cosmological constant”). See
the papers by M. Anderson in Notices of the AMS, 04.
3.5
Laplace(-Bertrami) operator
A bit of multilinear algebra
Let V be a real vector space with a positive definite inner product. Recall that
Ap (V ) is the space of alternating p-forms, and if {ω1 , . . . , ωn } is a basis of V ∗ ,
38
Differential Geometry
then {ωi1 ∧ · · · ∧ ωip | i1 < · · · < i p } is a basis of Ap (V ). We may also think of
Ap (V ) = ∧ p (V ∗ ), the pth exterior power of V ∗ .
Let {e1 , . . . , en } be an orthonormal basis of V with corresponding dual basis
{ω1 , . . . , ωn } of V ∗ . This induces an inner product on V ∗ with respect to which
{ω1 , . . . , ωn } is orthonormal. This induces also an inner product in Ap (V ) by
declaring the basis {ωi1 ∧ · · · ∧ ωip | i1 < · · · < i p } to be orthonormal. (Check
that 〈t 1 ∧ · · · ∧ t p , s1 ∧ · · · ∧ s p 〉 = det〈t i , s j 〉.) If V is oriented then we have made
a choice of a non-zero top form in An (V ). But this space is one dimensional, so
there is a unique ω g that has norm 1 and defined the orientation of V .
3.5.1 Definition. The Hodge ∗ operator is a linear operator
∗ : Ap (V ) → An−p (V ) : ωi1 ∧ · · · ∧ ωip 7→ ω j1 ∧ · · · ∧ ω jn−p
such that {ωi1 , . . . , ωip , ω j1 , . . . , ω jn−p } is a positively oriented basis of An (V ), e.g.
∗(ω1 ∧ · · · ∧ ω p ) = ω p+1 ∧ · · · ∧ ωn .
It’s easy to see (check) that for α, β ∈ Ap (V ), α ∧ ∗β = 〈α, β〉ω g . Also check
that ∗∗ = (−1) p(n−p) .
Back to manifolds
Let (M n , g) be an oriented Riemannian manifold. We can do the constructions
above over each Tx M and Tx∗ M and we get a volume form ω g (x) for each x ∈ M ,
called the Riemannian volume form. We also get a Hodge ∗ operator ∗ : Ω p (M ) →
Ωn−p (M ) such that α ∧ ∗β = 〈α, β〉 x ω g (x). Recall that d : Ω p (M ) → Ω p+1 (M ).
3.5.2 Definition. Define δ : Ω p (M ) → Ω p−1 (M ) by δ := (−1)n(p+1)+1 ∗ d∗ (think
of divergence). The Laplace operator on Ω p (M ) is ∆ : Ω p (M ) → Ω p (M ), defined
by ∆ := dδ + δd.
(elliptic PDE)
3.5.3 Proposition. Suppose that M is compact. Then
Z
Z
〈dα, β〉ω g =
M
〈α, δβ〉ω g
M
for all α ∈ Ω p−1 (M ) and β ∈ Ω p (M ).
PROOF: We have
d(α ∧ ∗β) = dα ∧ ∗β + (−1) p−1 α ∧ d(∗β)
= dα ∧ ∗β + (−1) p−1 α ∧ (−1)(p−1)(n−p+1) ∗ ∗d(∗β)
= dα ∧ ∗β − α ∧ ∗(δβ)
Z
Z
Z
d(α ∧ ∗β) =
M
0=
α ∧ ∗(δβ)
dα ∧ ∗β −
M
Z
M
〈dα, β〉ω g −
M
Z
〈α, δβ〉ω g
M
Laplace(-Bertrami) operator
39
by Stoke’s Theorem.
ƒ
3.5.3 means that for α, β ∈ Ω p (M ), with respect to the inner product
Z
(α, β) L 2 :=
〈α, β〉ω g ,
M
δ is the adjoint of d, i.e. (dα, β) L 2 = (α, δβ) L 2 . Notice that ∆ is self-adjoint with
respect to this inner product.
3.5.4 Definition. A form ω ∈ Ω p (M ) is harmonic if ∆ω = 0.
3.5.5 Corollary. ω is harmonic if and only if dω = 0 and δω = 0 (the latter
condition is referred to as co-closed).
PROOF: One direction is obvious. Suppose that ∆ω = 0, so
0 = (∆ω, ω) L 2 = ((dδ+δd)ω, ω) L 2 = (dδω, ω)+(δdω, ω) = (δω, δω)+(dω, dω).
Therefore both terms on the right hand side are zero since they are non-negative,
and ω is both closed and co-closed since the L 2 inner product is positive definite.ƒ
p
Recall that H dR (M , R) is the space of closed p-forms modulo the exact forms,
p
so a class a ∈ H dR (M , R) is of the form a = {α+ dβ | dα = 0, β ∈ Ω p−1 }. Let H p =
p
ker ∆, the space of harmonic p-forms. There is a natural map H p → H dR (M , R),
which turns out to be an isomorphism.
Locally, let (U, ϕ) to be a positively oriented chart inducing local coordinates
{ ∂ ∂x 1 , . . . , ∂ ∂x n } and {d x 1 , . . . , d x n }. Then
|d x 1 ∧ · · · ∧ d x n |2 = det(〈d x i d x j 〉) = det(g i j ),
so
p
p
det(g i j )ω g
and
ω g = det(g i j )d x 1 ∧ · · · ∧ d x n .
p
p
p
(In classical terms, ω g = EG − F 2 d x 1 ∧ d x 2 .) We write g := det(g i j ).
In the special case ∆ : Ω0 → Ω0 , ∆ f = dδ f + δd f = δd f . Suppose that ψ is
a function with support contained in U. Then
Z
Z
1 ∂
∂ f ∂ψp
p ij ∂ f
p
1
n
− p
gg
ψ g d x · · · d x = gi j
g d x1 · · · d xn
g ∂ xj
∂ xi
∂ xi ∂ x j
Z
d x1 ∧ · · · ∧ d xn =
=
〈d f , dψ〉ω g
= (δd f , ψ)
= (∆ f , ψ)
Z
=
=
∆ f ψω g
Z
p
∆ f ψ g d x 1 ∧ · · · ∧ d x n,
40
so ∆ f = − p1g
Differential Geometry
∂
∂ xj
p
Laplacian ∆ f = −
P
∂f
g g i j ∂ x i . With M = Rn and g i j = δi j , we recover the usual
∂2f
(∂ x j )2
(sometimes the minus sign is omitted in the literature).
3.5.6 Theorem (Hodge Decomposition Theorem, 1935).
Let M be a compact oriented manifold. Then dim H p < ∞ for every 0 ≤ p ≤ n,
and
Ω p (M ) = ∆Ω p (M ) ⊕ H p
= dδΩ p (M ) ⊕ δdΩ p (M ) ⊕ H p
= dΩ p−1 (M ) ⊕ δΩ p+1 (M ) ⊕ H p
and the direct sums are L 2 -orthogonal.
Remark. The first equality implies the other two. For orthogonality,
(dδβ, δdβ) = (d 2 δβ, dβ) = 0
since d 2 ≡ 0.
3.5.7 Corollary. Every de Rham cohomology class can be uniquely represented
by a harmonic form.
p
PROOF: For a class a ∈ H dR (M , R), a = [α] for some closed p-form α. By the
Hodge Decomposition Theorem, we may write α = dβ + δν + ω, where ω ∈ H p ,
β ∈ Ω p−1 , and ν ∈ Ω p+1 . Then 0 = dα = dδν, so 0 = (dδν, ν) = (δν, δν),
implying δν = 0. Therefore α = ω + dβ, so [ω] = [α] = a.
If ω1 and ω2 were two harmonic p-forms giving the same class then ω1 −ω2 =
dβ for some β ∈ Ω p−1 , and δdβ = δ(ω1 − ω2 ) = 0, so (dβ, dβ) = (δdβ, β) = 0,
implying dβ = 0 and ω1 = ω2 .
ƒ
p
Therefore H p → H dR (M , R) : ω 7→ [ω] is a linear isomorphism. Consequently
dim H p (M , R) < ∞ for all p when M is compact (and this is not an a priori trivial
fact). These dimensions are called the Betti numbers.
Aside: Write
Z
Z
1 2
1
1
e(α) := |α| L 2 =
〈α, α〉ω g =
α ∧ α∗ ,
2
2
2
the energy of α. If α is closed then the cohomology class of α is represented by all
forms of the form α + dβ. Write e(t) = 21 |α + t dβ|2L 2 , so
e(t) =
Š
1€
(α, α) + 2t(α, dβ) + t 2 (dβ, dβ)
2
and
e0 (0) = (α, dβ) = (δα, β).
Then e0 (0) = 0 for all β if and only if δα = 0, so the harmonic p-forms are the
critical points of e on the cohomology class.
Laplace(-Bertrami) operator
41
Poincaré Duality
R
Define a pairing Ω p (M )n−p (M ) → R : (α, β) 7→ M α∧β. We claim that the pairing
R
descends to cohomology ([α], [β]) 7→ M α ∧ β. But
Z
(α + dν) ∧ β =
M
Z
α∧β +
M
Z
(dν) ∧ β =
M
Z
d(ν ∧ β) = 0
M
by Stoke’s Theorem, since d(ν ∧β) = (dν)∧β ±ν ∧ dβ and β is closed. We further
claim that the pairing H p (M , R) × H n−p (M , R) → R is non-degenerate. Indeed, if
α is a harmonic p-form then β = ∗α is a harmonic n − p form (check), so
Z
([α], [β]) 7→
α ∧ ∗α = |α|2L 2 6= 0
when α 6= 0. Since each class has a harmonic representative, this shows the forms
is non-degenerate. It follows that when M is compact and connected, H n (M , R) =
H 0 (M , R) = R.
Outline of the proof of the Hodge decomposition theorem
“A foray into the dungeons of the analysts. They are one who make the world tick.
If you spend enough time down there you might like it. Let’s hope they don’t turn
out the lights. . . ”
We would like to show that Ω p = ∆Ω p ⊕ H p , and this direct sum is L 2 orthogonal. In some sense we need to solve the equation ∆ω = α. Notice that for
any ϕ, 〈∆ω, ϕ〉 = 〈α, ϕ〉, so (∆ω, ϕ) L 2 = (α, ϕ) L 2 , or (ω, ∆ϕ) L 2 = (α, ϕ) L 2 since
∆ is self-adjoint.
3.5.8 Definition. A linear functional ` : Ω p → R is said to be a weak solution of
∆ω = α if
(i) ` is bounded, i.e. |`(β)| ≤ C|β| for some C constant and all β ∈ Ω p .
(ii) `(∆ϕ) = (α, ϕ) L 2 for all ϕ ∈ Ω p .
If ω is an honest solution then `ω (ϕ) = (ω, ϕ) L 2 is a weak solution. Can we
find even weak solutions? If we can one, can we get an honest solution out of it?
3.5.9 Theorem (Regularity Theorem). Any weak solution of ∆ω = α is of the
form `ω for some ω ∈ Ω p .
3.5.10 Theorem (Compactness Theorem). Let αn ∈ Ω p , n ≥ 1. If there is a
constant C such that |αn |, |∆αn | < C for all n then αn has a Cauchy subsequence.
First note that the Compactness Theorem implies that H p is finite dimensional. Indeed, if not then there would exist an infinite orthonormal sequence
of orthonormal harmonic forms, which would contradict Compactness. Therefore
42
Differential Geometry
we may write Ω p = H p ⊕ (H p )⊥ . Note that ∆Ω p ⊆ (H p )⊥ , since (∆ϕ, ω) =
(ϕ, ∆ω) = 0 when ω ∈ H p . It remains to prove that given α ∈ (H p )⊥ we can
find ω ∈ Ω p such that ∆ω = α. To find a weak solution, proceed as follows.
Define `|∆Ωp as `(∆ϕ) = (ϕ, α) L 2 . This is well-defined since for ϕ1 , ϕ2 ∈ Ω p with
∆ϕ1 = ∆ϕ2 , ϕ1 − ϕ2 ∈ H p , so (ϕ1 , α) = (ϕ2 , α) for all α ∈ ϕ1 − ϕ2 ∈ (H p )⊥ . We
must also check that `|∆Ωp is bounded (one can prove this again using the Compactness Theorem). The Hahn-Banach theorem allows us to extend ` to all of Ω p ,
and the Regularity Theorem gives us an honest solution.
∆ is an elliptic operator. Elliptic operators have finite dimensional spaces of
solutions and the Regularity Theorem holds for them.
Ellipticity
We will work in Rn for the moment. Consider a function
u : Rn → R and a
P
general linear partial differential operator (PDO) P = |α|≤k aα (x)∂ α , where α =
P
|α|u
(α1 , . . . , αn ) ∈ Nn , |α| = i αi , and ∂ α = ∂ x α∂1 ···∂ x αn . The “symbol of P” is given by
n
1
P
Q α
replacing ∂∂x j with iξ j , so P(x, ξ) = |α|≤k aα (x)(iξ)α , where y α = j y j j . The
P
Pn
2
“principal symbol” is pk (x, ξ) = |α|=k aα (x)(iξ)α . For example, if P = i=1 ∂∂x 2
then P2 (x, ξ) = −|ξ|2 . Associated with the “wave operator”
∂ 2u
∂ x 12
=
∂ 2u
∂ x 22
i
the symbol
is is −ξ21 + ξ22 . P is elliptic at x if Pk (x, ξ) 6= 0 for all ξ 6= 0. P is elliptic it is elliptic
at all x.
Recall that
∂f
1 ∂
i jp
g
∆f = −p
g
.
g ∂ xj
∂ xi
∂2f
The higher order term is g i j ∂ x i ∂ x j . (g i j ) is positive semi-definite, so ∆ is elliptic
(under an appropriate extension of the definition of ellipticity to vector bundles
over manifolds).
3.5.11 Theorem (Bochner, 1946). Let M be a compact, orientable, connected
Riemannian manifold. Suppose that Ric ≥ 0 (i.e. this symmetric, bilinear form is
positive semi-definite). Then every harmonic 1-form is parallel.
For ω ∈ H 1 , ω x (v) = 〈X (x), v〉, where X is the dual vector field. ω is parallel
if and only if ∇X = 0. If ω x = 0 then ω ≡ 0 (since ω is parallel), so H 1 → Tx∗ M :
ω 7→ ω x is an injective linear map. Therefore dim H 1 ≤ dim M . In particular,
dim H 1 (M , R) ≤ dim M .
Suppose Σ g is a surface of genus g. Then Σ g × S n (n ≥ 2) is an (n + 2)dimensional manifold, so if there is a positive semi-definite Ricci curvature then
2g = dim H 1 ≤ n + 2. For large enough g we get a family of neat examples.
PROOF (OF B OCHNER’S THEOREM): If ω is a harmonic 1-form then the following
identity holds for the dual vector field X .
−∆( 12 |X |2 ) = |∇X |2 + Ric(X , X ).
Yang-Mills Equations
43
Integrating over M yields
0=
(−∆( 12 |X |2 ), 1) L 2
=
Z
−∆( 12 |X |2 )ω g
=
Z
|∇X | ω g +
2
Z
Ric(X , X )ω g .
Whence ∇X = 0 since Ric(X , X ) ≥ 0.
ƒ
This ends the examinable material of the course.
3.6
Yang-Mills Equations
Let M n be a compact oriented manifold and E a vector bundle of rank k over
M . Suppose that we have a Riemannian metric on M and a Riemannian metric
on E, and a metric compatible connection D, i.e. for s1 , s2 ∈ Γ(E) = Ω0M (E) and
X ∈ V (M ),
X 〈s1 , s2 〉 = DX s1 , s2 〉 + 〈s1 , DX s2 〉,
also called an orthogonal connection. In an orthogonal trivialization D = d + A,
and A(X ) is a skew-symmetric matrix. Consider the structure group O(k), and
notice that A(X ) ∈ o(k), the Lie algebra of O(k). If F D is the curvature of D
p
then F D ∈ Ω2M (End E), and F D is also skew-symmetric. Recall that Ω M (End E) =
p
p
Γ(A (M ) ⊗ End E) is the collection of End E valued p-forms. Let Ω M (Ad E) be the
Ad(E)-valued p-forms, where Ad E is the follection of bundles over M given by
elements in End E which are skew-symmetric. E.g. F D ∈ Ω2M (Ad E), and if D1 and
D2 are two metric compatible connections then D1 − D2 ∈ Ω1M (Ad E).
p
We will now define an L 2 -inner product on Ω M (Ad E) = Γ(Ap (M ) ⊗ Ad E). If A
and B are skew-symmetric matrices then A· B = − tr(AB) is a natural inner product
(it is the Killing form on o(k)). For ω ∈ Ω p (M ) and s ∈ Γ(Ad E), define
〈ω1 ⊗ s1 , ω2 ⊗ s2 〉 x = − tr(s1 s2 )〈ω1 , ω2 〉 x
and as before define
(ω1 ⊗ s1 , ω2 ⊗ s2 ) L 2 =
Z
〈ω1 ⊗ s1 , ω2 ⊗ s2 〉 x ω g
M
p
n−p
We also have a ∗-operator acting Ω M (Ad E) → Ω M (Ad E) as ω ⊗ s 7→ (∗ω) ⊗ s.
3.6.1 Definition. The Yang-Mills functional is
Y M : D → (F D , F D ) L 2 =
Z
〈F D , F D 〉 x ω g .
M
44
Differential Geometry
Harmonic forms pop up as critical points of the energy function. What are the
critical points of the Yang-Mills functional? Let A ∈ Ω1M (Ad E), so we have
F D+tA(σ) = (D + tA)((D + tA)(σ))
= (D + tA)(Dσ + tA)(σ)
= D2 σ + t D(Aσ) + tA(Dσ) + t 2 A ∧ A(σ)
= D2 σ + t((DA)σ − A ∧ Dσ) + tA(Dσ) + t 2 A ∧ A(σ)
= (F D + t DA + t 2 (A ∧ A))σ)
p
p+1
since we can extend D : Ω M (Ad E) → Ω M (Ad E) to satisfy a Leibniz rule as we
have earlier in the course. Therefore
d Y M (D + tA) = 2(DA, F D ) L 2 .
d t t=0
p
p+1
p
p−1
For D : Ω M (Ad E) → Ω M (Ad E), define D∗ : Ω M (Ad E) → Ω M (Ad E) to be the
L 2 -adjointof D. Then critical points of the Yang-Mills functional are exactly those
D for which D∗ F D = 0. This is the Yang-Mills equation. Recall also the Bianchi
Identity which gives DF D = 0.
In the case p = 2, ∗∗ = (−1)2(n−2) = 1. One may check that when the dimension of M is even then D∗ = − ∗ D∗. Whence in this case the Yang-Mills equation
reduces to D ∗ F D = 0.
Remark.
(i) The same construction can be done (and often is in Physics) for a Hermitian
inner product and unitary connections.
(ii) When k = 2 in the O(2) orthogonal case or k = 1 in the U(1) unitary case,
we get Maxwell’s equations. We get Hodge theory for 2-forms (D ∗ F D = 0
corresponds to F D being co-closed and DF D = 0 corresponds to F D being
closed).
Locally, if g i j = δi j (say if M = R4 ) then if D = d + A, where A = Ai d x 1 and
F D = Fi j d x i ∧ d x j , then
‚
Œ
∂ Fi j
∗
D FD = −
− [Ai , Fi j ] d x j = 0
∂ xi
so the Yang-Mills equations are
δFi j
∂ xi
δA
δA
+ [Ai , Fi j ] = 0for j = 1, . . . , n
where Fi j = ∂ x ji − ∂ x ij + [Ai , A j ].
Again, the set up is as follows. E → M is a vector bundle, we have Riemannian metrics on E and M , and a metic/compatible/orthogonal connection D. We
p
p+1
extend D to Ω M (Ad E) → Ω M (Ad E).
Yang-Mills Equations
45
3.6.2 Definition. The Gauge group is g = AutG (E), the collection of automorphisms h of E such that for each x ∈ M , h x : E x → E x is orthogonal, where G is
the structure group (O(k) in our case).
The important point is that g acts on the space of metric connections. If h ∈
AutG (E) and D is a metric connection then define h∗ D = h−1 ◦ D ◦ h, i.e. h∗ D(s) =
h−1 D(hs) for all s ∈ Γ(E). Locally one may check that if D = d + A then h∗ (A) =
h−1 dh + h−1 Ah and h∗ F = h−1 ◦ F ◦ h. Further,
Y M (h∗ D) = (Fh∗ D , Fh∗ D ) L 2 = (F D , F D ) L 2 = Y M (D)
since each element of AutG (E) is an isometry. Hence Y M is invariant under the
Gauge group. It follows that critcal points, i.e. Yang-Mills connections, are mapped
to one another under the action of g.
3.6.3 Definition. The moduli space M of M is the space of Yang-Mills connections
modulo the action of g.
3.6.4 Example. When k = 2 and E = M × R2 is the trival rank 2 vector bundle
over a compact oriented manifold Riemannian manifold M , the structure group
is SO(2). The associated Lie algebra is skew-symmetric 2 × 2 matrices and so is
isomorphic to R. Then we may
” write
—D = d + A globally since E is trivial, where
0 a
A ∈ Ω1M (Ad E). Then A(x) = −a x 0x where a x is a 1-form, so we may think of
1
1
A as an ordinary
0 a 1-form,
s1 and Ω M (Ad E) = Ω (M ). For s ∈ Γ(E), s : M → R, so
Ds = ds + −a 0
s2 , and
0
da
.
F D = dA + A ∧ A = dA =
−da 0
The Yang-Mills equation gives that δda = 0, so da is a harmonic 2-form. By
triviality, δd a = 0 and d a harmonic imply that da = 0, so a is a closed 1-form.
Hence any Yang-Mills connection is given by a closed 1-form on M , and visa versa.
What is the action of g = SO(2)? For x ∈ M , h(x) ∈ SO(2) and h∗ (A) =
−1
0 u
h dh + h−1 Ah, and h(x) is orthogonal, so we may write h(x) = exp −u
0 . Thus
∗
h (A) = du + A since SO(2) is Abelian. As we let u run over R, we see that
M = H 1 (M , R). In particular, if H 1 (M , R) = {0} then M is just a single point.
3.6.5 Theorem (Donaldson, mid 80’s). Let M be a compact simply connected
4-manifold. We have a pairing
Z
Γ : H 2 (M , R) × H 2 (M , R) → R : ([α], [β]) 7→
α ∧ β,
and if Γ is definite then there is a basis such that it is ±I.
R
Consider in particular an SU(2)-bundle over M with h(F ∧ F ) = −8π2 . He
proved that in this case, M is compact, 5-dimensional, and is smooth a.e. except
for some “core singularities” and has M as a boundry. And so on. . .
Index
Ap (V ), 13
C ∞ differentiable structure, 2
T M, 8
T∗M, 8
Tp M , 6
Γ(E), 29
Ω p (M ), 14
Einstein manifold, 37
elliptic, 42
elliptic operator, 42
embedded submanifold, 11
energy, 40
exact, 16
exponential, 4
exterior product, 13
associated bundles, 25
fibre, 8
first order linear operator, 9
flat connection, 32
flow, 8
frame, 31
free action, 24
base space, 23
Betti numbers, 40
boundry, 19
bundle automorphism, 27
bundle chart, 8
bundle isomorphism, 27
bundle morphism, 26
Gauge group, 45
geodesic, 23
canonical 1-form, 22
canonical basis, 7
canonical submersion, 12
chart, 2
closed, 16
co-closed, 39
cocyle conditions, 24
codimension zero submanifold, 19
coefficients of the connection, 28
compatible, 2
connection, 28, 29, 34
continuous map, 2
coordinate neighbourhood, 2
cotangent bundle, 8
covariant-constant, 32
curvature, 32
half-translation, 10
Hamiltonian flow, 23
Hamiltonian vector field, 23
harmonic, 39
Haudorff, 2
Hodge ∗ operator, 38
homeomorphism, 2
Hopf bundle, 26
Hopf map, 26
horizontal subspace, 27
immersion, 11, 12
integral curves, 8
Laplace operator, 38
left-invariant vector field, 11
Legendre transform, 21
Levi-Civita, 31
Levi-Civita connection, 35
Lie algebra, 10
Lie bracket, 9
Lie group, 4
local diffeomorphism, 12
local operator, 30
logarithm, 4
de Rham cohomology, 16
de Rham complex, 16
derivation, 7
diffeomorphic, 3
diffeomorphism, 3
differential, 6
differential p-form, 14
dimension, 3
domain with smooth boundry, 19
dual basis, 14
47
48
metric connection, 33
moduli space, 45
musical isomorphisms, 21
open, 2
orientable, 17
orientation preserving, 18
orthogonal connection, 33
orthogonal trivialization, 34
outward directed, 19
parallel, 32
parallel transform, 32
parallelizable, 8
partition of unity, 17
phase space, 8
positively oriented, 19
principal bundle, 24
product connection, 33
pullback, 14, 15
pullback bundle, 26
rank, 23
regular value, 12
Ricci curvature, 37
Ricci flat, 37
Ricci flow, 37
Riemannian manifold, 21
Riemannian metric, 21
Riemannian volume form, 38
scalar curvature, 37
second countable, 2
section, 8, 25
sectional curvature, 37
smooth 1-form, 8
smooth action, 24
smooth map, 3
smooth structure, 2
structure group, 24
submersion, 12
subordinate, 17
symmetric, 34
symplectic form, 21
symplectic manifold, 21
tangent bundle, 8
INDEX
tangent space, 6
tautological bundle, 25
topological space, 2
torsion, 34
total space, 23
transition function, 24, 25
trivial bundle, 24
trivial connection, 33
trivialization, 23
trivializing neighbourhood, 23
typical fibre, 23
unitary connection, 33
vector bundle, 8, 23
vector field, 8
vertical subspace, 27
volume form, 14
weak solution, 41
wedge product, 13, 15
Yang-Mills equation, 44
Yang-Mills functional, 43