<<

. 4
( 5)



>>

t t’1
&%




2. Let (P2 , ωFS ) be 2-(complex-)dimensional complex projective space equipped
with the Fubini-Study form de¬ned in Section 5.1. The T2 -action on P2 by
(eiθ1 , eiθ2 ) · [z0 : z1 : z2 ] = [z0 : eiθ1 z1 : eiθ2 z2 ] has moment map


|z1 |2 |z2 |2
1
µ[z0 : z1 : z2 ] = ’ , .
|z0 |2 + |z1 |2 + |z2 |2 |z0 |2 + |z1 |2 + |z2 |2
2
81
5.8. SYMPLECTIC TORIC MANIFOLDS



T



(’ 1 , 0)
t(0, 0) E
2
t
d     
 
d    
 
d   
 
d  
 

dt
(0, ’ 1 )
2




The ¬xed points get mapped as

[1 : 0 : 0] ’’ (0, 0)
1
[0 : 1 : 0] ’’ ’ 2 , 0
1
[0 : 0 : 1] ’’ 0, ’ 2 .

Notice that the stabilizer of a preimage of the edges is S 1 , while the action
is free at preimages of interior points of the moment polytope.


Exercise 32
Compute a moment polytope for the T 3-action on P 3 as
(eiθ1 , eiθ2 , eiθ3 ) · [z0 : z1 : z2 : z3 ] = [z0 : eiθ1 z1 : eiθ2 z2 : eiθ3 z3 ] .


Exercise 33
Compute a moment polytope for the T 2-action on P 1 — P 1 as
(eiθ , ei· ) · ([z0 : z1 ], [w0 : w1 ]) = ([z0 : eiθ z1 ], [w0 : ei· w1 ]) .


By Proposition 4.17, symplectic toric manifolds are optimal hamiltonian
torus-spaces. By Theorem 4.15, they have an associated polytope. It turns out
that the moment polytope contains enough information to sort all symplectic toric
manifolds. We now de¬ne the class of polytopes which arise in the classi¬cation.

De¬nition 5.11 A Delzant polytope ∆ in Rn is a polytope satisfying:
• simplicity, i.e., there are n edges meeting at each vertex;
• rationality, i.e., the edges meeting at the vertex p are rational in the sense
that each edge is of the form p + tui , t ≥ 0, where ui ∈ Z n;
82 LECTURE 5. SYMPLECTIC REDUCTION

• smoothness, i.e., for each vertex, the corresponding u 1 , . . . , un can be cho-
sen to be a Z-basis of Z n.


Examples of Delzant polytopes in R2 :


d d
d d
d d
d d


The dotted vertical line in the trapezoidal example is there just to stress that it is a
picture of a rectangle plus an isosceles triangle. For “taller” triangles, smoothness
would be violated. “Wider” triangles (with integral slope) may still be Delzant.
The family of the Delzant trapezoids of this type, starting with the rectangle,
correspond, under the Delzant construction, to the so-called Hirzebruch surfaces.
™¦
Examples of polytopes which are not Delzant:



d
rr „d
rr „d
r
rr „d
r „ 


The picture on the left fails the smoothness condition, since the triangle is not
™¦
isosceles, whereas the one on the right fails the simplicity condition.

Delzant™s theorem classi¬es (equivalence classes of) symplectic toric manifolds
in terms of the combinatorial data encoded by a Delzant polytope.

Theorem 5.12 (Delzant [14]) Toric manifolds are classi¬ed by Delzant poly-
topes. More speci¬cally, the bijective correspondence between these two sets is given
by the moment map:

{toric manifolds} ←’ {Delzant polytopes}
(M 2n , ω, Tn , µ) ’’ µ(M ) .

In Section 5.9 we describe the construction which proves the (easier) existence
part, or surjectivity, in Delzant™s theorem. In order to prepare that, we will next
give an algebraic description of Delzant polytopes.
83
5.9. DELZANT™S CONSTRUCTION

Let ∆ be a Delzant polytope in (Rn )—1 and with d facets.2 Let vi ∈ Z n,
i = 1, . . . , d, be the primitive3 outward-pointing normal vectors to the facets of ∆.
Then we can describe ∆ as an intersection of halfspaces
∆ = {x ∈ (Rn )— | x, vi ¤ »i , i = 1, . . . , d} for some »i ∈ R .

Example. For the picture below, we have
= {x ∈ (R2 )— | x1 ≥ 0, x2 ≥ 0, x1 + x2 ¤ 1}

= {x ∈ (R2 )— | x, (’1, 0) ¤ 0 , x, (0, ’1) ¤ 0 , x, (1, 1) ¤ 1} .
™¦



 
 
 
(0, 1)
  ’’ ’’
r
d   v3 = (1, 1)
d  
d  
' d 
’’’
’’ d
v2 = (’1, 0)
d
d
dr
r
(0, 0) (1, 0)



’’’
’’
v1 = (0, ’1)
c




5.9 Delzant™s Construction
Following [14, 24], we prove the existence part (or surjectivity) in Delzant™s the-
orem, by using symplectic reduction to associate to an n-dimensional Delzant
polytope ∆ a symplectic toric manifold (M∆ , ω∆ , Tn , µ∆ ).
1 Although we identify Rn with its dual via the euclidean inner product, it may be more clear
to see ∆ in (Rn )— for Delzant™s construction.
2 A face of a polytope ∆ is a set of the form F = P © {x ∈ R n | f (x) = c} where c ∈ R

and f ∈ (Rn )— satis¬es f (x) ≥ c, ∀x ∈ P . A facet of an n-dimensional polytope is an (n ’ 1)-
dimensional face.
3 A lattice vector v ∈ Z n is primitive if it cannot be written as v = ku with u ∈ Z n, k ∈ Z

and |k| > 1; for instance, (1, 1), (4, 3), (1, 0) are primitive, but (2, 2), (4, 6) are not.
84 LECTURE 5. SYMPLECTIC REDUCTION

Let ∆ be a Delzant polytope with d facets. Let vi ∈ Z n, i = 1, . . . , d, be the
primitive outward-pointing normal vectors to the facets. For some »i ∈ R, we can
write
∆ = {x ∈ (Rn )— | x, vi ¤ »i , i = 1, . . . , d} .
Let e1 = (1, 0, . . . , 0), . . . , ed = (0, . . . , 0, 1) be the standard basis of Rd . Consider

Rd ’’ Rn
π:
’’ vi .
ei

Lemma 5.13 The map π is onto and maps Z d onto Z n.

Proof. The set {e1 , . . . , ed } is a basis of Z d. The set {v1 , . . . , vd } spans Z n for the
following reason. At a vertex p, the edge vectors u1 , . . . , un ∈ (Rn )— , form a basis
for (Z n)— which, by a change of basis if necessary, we may assume is the standard
basis. Then the corresponding primitive normal vectors to the facets meeting at
p are symmetric (in the sense of multiplication by ’1) to the ui ™s, hence form a
basis of Z n.
Therefore, π induces a surjective map, still called π, between tori:
π
Rd /(2πZ d) ’’ Rn /(2πZ n)

Td Tn
’’ ’’ 1 .

The kernel N of π is a (d ’ n)-dimensional Lie subgroup of T d with inclusion
i : N ’ Td . Let n be the Lie algebra of N . The exact sequence of tori
i π
1 ’’ N ’’ Td ’’ Tn ’’ 1

induces an exact sequence of Lie algebras
i π
0 ’’ n ’’ Rd ’’ Rn ’’ 0

with dual exact sequence
π— i—
0 ’’ (Rn )— ’’ (Rd )— ’’ n— ’’ 0 .
i
Now consider Cd with symplectic form ω0 = dzk § d¯k , and standard
z
2
hamiltonian action of Td given by

(eit1 , . . . , eitd ) · (z1 , . . . , zd ) = (eit1 z1 , . . . , eitd zd ) .

The moment map is φ : Cd ’’ (Rd )— de¬ned by
1
φ(z1 , . . . , zd ) = ’ (|z1 |2 , . . . , |zd |2 ) + constant ,
2
85
5.9. DELZANT™S CONSTRUCTION

where we choose the constant to be (»1 , . . . , »d ). The subtorus N acts on Cd in a
hamiltonian way with moment map

i— —¦ φ : Cd ’’ n— .

Let Z = (i— —¦ φ)’1 (0) be the zero-level set.
Claim 1. The set Z is compact and N acts freely on Z.
We postpone the proof of this claim until further down.
Since i— is surjective, 0 ∈ n— is a regular value of i— —¦ φ. Hence, Z is a
compact submanifold of Cd of (real) dimension 2d ’ (d ’ n) = d + n. The orbit
space M∆ = Z/N is a compact manifold of (real) dimension dim Z ’ dim N =
(d + n) ’ (d ’ n) = 2n. The point-orbit map p : Z ’ M∆ is a principal N -bundle
over M∆ . Consider the diagram
j
’ Cd
Z
p“
M∆

where j : Z ’ Cd is inclusion. The Marsden-Weinstein-Meyer theorem guarantees
the existence of a symplectic form ω∆ on M∆ satisfying

p — ω∆ = j — ω0 .

Since Z is connected, the compact symplectic 2n-dimensional manifold (M∆ , ω∆ )
is also connected.
Proof of Claim 1. The set Z is clearly closed, hence in order to show that it is
compact it su¬ces (by the Heine-Borel theorem) to show that Z is bounded. Let
∆ be the image of ∆ by π — . We will show that φ(Z) = ∆ .

Lemma 5.14 Let y ∈ (Rd )— . Then:

y∈∆ ⇐’ y is in the image of Z by φ .

Proof. The value y is in the image of Z by φ if and only if both of the following
conditions hold:
1. y is in the image of φ;
2. i— y = 0.
Using the expression for φ and the third exact sequence, we see that these
conditions are equivalent to:

1. y, ei ¤ »i for i = 1, . . . , d;
86 LECTURE 5. SYMPLECTIC REDUCTION

2. y = π — (x) for some x ∈ (Rn )— .

Suppose that the second condition holds, so that y = π — (x). Then

π — (x), ei ¤ »i , ∀i
y, ei ¤ »i , ∀i ⇐’
⇐’ x, π(ei ) ¤ »i , ∀i
⇐’ x, vi ¤ »i , ∀i
⇐’ x∈∆.

Thus, y ∈ φ(Z) ⇐’ y ∈ π — (∆) = ∆ .
Since we have that ∆ is compact, that φ is a proper map and that φ(Z) = ∆ ,
we conclude that Z must be bounded, and hence compact.
It remains to show that N acts freely on Z.
Pick a vertex p of ∆, and let I = {i1 , . . . , in } be the set of indices for the n
facets meeting at p. Pick z ∈ Z such that φ(z) = π — (p). Then p is characterized
by n equations p, vi = »i where i ranges in I:

⇐’
p, vi = »i p, π(ei ) = »i
π — (p), ei = »i
⇐’
⇐’ φ(z), ei = »i
⇐’ i-th coordinate of φ(z) is equal to »i
1
’ 2 |zi |2 + »i = »i
⇐’
⇐’ zi = 0 .

Hence, those z™s are points whose coordinates in the set I are zero, and whose
other coordinates are nonzero. Without loss of generality, we may assume that
I = {1, . . . , n}. The stabilizer of z is

(Td )z = {(t1 , . . . , tn , 1, . . . , 1) ∈ Td } .

As the restriction π : (Rd )z ’ Rn maps the vectors e1 , . . . , en to a Z-basis
v1 , . . . , vn of Z n (respectively), at the level of groups, π : (Td )z ’ Tn must be
bijective. Since N = ker(π : Td ’ Tn ), we conclude that N © (Td )z = {e}, i.e.,
Nz = {e}. Hence all N -stabilizers at points mapping to vertices are trivial. But
this was the worst case, since other stabilizers Nz (z ∈ Z) are contained in sta-
bilizers for points z which map to vertices. This concludes the proof of Claim 1.

Given a Delzant polytope ∆, we have constructed a symplectic manifold
(M∆ , ω∆ ) where M∆ = Z/N is a compact 2n-dimensional manifold and ω∆ is the
reduced symplectic form.
Claim 2. The manifold (M∆ , ω∆ ) is a hamiltonian Tn -space with a moment map
µ∆ having image µ∆ (M∆ ) = ∆.
87
5.9. DELZANT™S CONSTRUCTION

Proof of Claim 2. Let z be such that φ(z) = π — (p) where p is a vertex of ∆, as
in the proof of Claim 1. Let σ : Tn ’ (Td )z be the inverse for the earlier bijection
π : (Td )z ’ Tn . Since we have found a section, i.e., a right inverse for π, in the
exact sequence
i π
’’ Td ’’ Tn
1 ’’ N ’’ 1 ,
σ
←’

the exact sequence splits, i.e., becomes like a sequence for a product, as we obtain
an isomorphism
(i, σ) : N — Tn ’’ Td .
The action of the Tn factor (or, more rigorously, σ(Tn ) ‚ Td ) descends to the
quotient M∆ = Z/N .
It remains to show that the Tn -action on M∆ is hamiltonian with appropriate
moment map.
Consider the diagram
j σ—
φ
’ Cd ’’ (Rd )— · — • (Rn )— ’’ (Rn )—
Z
p“
M∆

where the last horizontal map is simply projection onto the second factor. Since
the composition of the horizontal maps is constant along N -orbits, it descends to
a map
µ∆ : M∆ ’’ (Rn )—
which satis¬es
µ∆ —¦ p = σ — —¦ φ —¦ j .
By Section 5.5 on reduction for product groups, this is a moment map for the
action of Tn on (M∆ , ω∆ ). Finally, the image of µ∆ is:

µ∆ (M∆ ) = (µ∆ —¦ p)(Z) = (σ — —¦ φ —¦ j)(Z) = (σ — —¦ π — )(∆) = ∆ ,

because φ(Z) = π — (∆) and σ — —¦ π — = (π —¦ σ)— = id.
We conclude that (M∆ , ω∆ , Tn , µ∆ ) is the required toric manifold correspond-
ing to ∆.


Exercise 34
Let ∆ be an n-dimensional Delzant polytope, and let (M ∆ , ω∆ , T n, µ∆ ) be the
associated symplectic toric manifold. Show that µ ∆ maps the ¬xed points of
T n bijectively onto the vertices of ∆.
88 LECTURE 5. SYMPLECTIC REDUCTION

Exercise 35
Follow through the details of Delzant™s construction for the case of ∆ = [0, a] ‚
R— (n = 1, d = 2). Let v(= 1) be the standard basis vector in R. Then ∆ is
described by
x, ’v ¤ 0 and x, v ¤ a ,
where v1 = ’v, v2 = v, »1 = 0 and »2 = a.


v
ta
T



t0
c
’v


The projection
π
R2 ’’ R
e1 ’’ ’v
e2 ’’ v
has kernel equal to the span of (e1 + e2 ), so that N is the diagonal subgroup
of T 2 = S 1 — S 1 . The exact sequences become
i π
T2 S1
1 ’’ N ’’ ’’ ’’ 1
t ’’ (t, t)
t’1 t2
(t1 , t2 ) ’’ 1

i π
R2
0 ’’ ’’ ’’ ’’ 0
R
n
x ’’ (x, x)
(x1 , x2 ) ’’ x 2 ’ x1

π— i—
(R2 )—
R— n—
0 ’’ ’’ ’’ ’’ 0
x ’’ (’x, x)
(x1 , x2 ) ’’ x 1 + x2 .

The action of the diagonal subgroup N = {(eit , eit ) ∈ S 1 — S 1 } on C2 ,
(eit , eit ) · (z1 , z2 ) = (eit z1 , eit z2 ) ,
has moment map
1
(i— —¦ φ)(z1 , z2 ) = ’ 2 (|z1 |2 + |z2 |2 ) + a ,
with zero-level set
(i— —¦ φ)’1 (0) = {(z1 , z2 ) ∈ C2 : |z1 |2 + |z2 |2 = 2a} .
Hence, the reduced space is a projective space:
(i— —¦ φ)’1 (0)/N = P 1 .


Example. Consider
(S 2 , ω = dθ § dh, S 1 , µ = h) ,
89
5.9. DELZANT™S CONSTRUCTION

where S 1 acts on S 2 by rotation. The image of µ is the line segment I = [’1, 1].
The product S 1 — I is an open-ended cylinder. By collapsing each end of the
™¦
cylinder to a point, we recover the 2-sphere.

Exercise 36
Build P 2 from T 2 — ∆ where ∆ is a right-angled isosceles triangle.


Exercise 37
Consider the standard (S 1 )3 -action on P 3:
(eiθ1 , eiθ2 , eiθ3 ) · [z0 : z1 : z2 : z3 ] = [z0 : eiθ1 z1 : eiθ2 z2 : eiθ3 z3 ] .
Exhibit explicitly the subsets of P 3 for which the stabilizer under this action
is {1}, S 1 , (S 1 )2 and (S 1 )3 . Show that the images of these subsets under the
moment map are the interior, the facets, the edges and the vertices, respec-
tively.


Exercise 38
What would be the classi¬cation of symplectic toric manifolds if, instead of
the equivalence relation de¬ned in Section 5.8, one considered to be equivalent
those (Mi , ωi , T i, µi ), i = 1, 2, related by an isomorphism » : T 1 ’ T 2 and a
»-equivariant symplectomorphism • : M 1 ’ M2 such that:
(a) the maps µ1 and µ2 —¦ • are equal up to a constant?
(b) we have µ1 = —¦ µ2 —¦ • for some ∈ SL(n; Z)?

Exercise 39

(a) Classify all 2-dimensional Delzant polytopes with 3 vertices, i.e., trian-
gles, up to translation, change of scale and the action of SL(2; Z).
Hint: By a linear transformation in SL(2; Z), we can make one of the angles in
the polytope into a square angle. How are the lengths of the two edges forming
that angle related?

(b) Classify all 2-dimensional Delzant polytopes with 4 vertices, up to trans-
lation and the action of SL(2; Z).
Hint: By a linear transformation in SL(2; Z), we can make one of the angles
in the polytope into a square angle. Check that automatically another angle
also becomes 90o .

(c) What are all the 4-dimensional symplectic toric manifolds that have four
¬xed points?

Exercise 40
Let ∆ be the n-simplex in Rn spanned by the origin and the standard basis
vectors (1, 0, . . . , 0), . . . , (0, . . . , 0, 1). Show that the corresponding symplectic
toric manifold is projective space, M ∆ = P n.


Exercise 41
Which 2n-dimensional toric manifolds have exactly n + 1 ¬xed points?
Appendix A

Prerequisites from
Di¬erential Geometry

A.1 Isotopies and Vector Fields
Let M be a manifold, and ρ : M — R ’ M a map, where we set ρt (p) := ρ(p, t).
De¬nition A.1 The map ρ is an isotopy if each ρt : M ’ M is a di¬eomor-
phism, and ρ0 = idM .
Given an isotopy ρ, we obtain a time-dependent vector ¬eld, that is, a
family of vector ¬elds vt , t ∈ R, which at p ∈ M satisfy
d
q = ρ’1 (p) ,
vt (p) = ρs (q) where t
ds s=t
i.e.,
dρt
= vt —¦ ρt .
dt
Conversely, given a time-dependent vector ¬eld vt , if M is compact or if the
vt ™s are compactly supported, there exists an isotopy ρ satisfying the previous
ordinary di¬erential equation.
Suppose that M is compact. Then we have a one-to-one correspondence
{isotopies of M } ←’ {time-dependent vector ¬elds on M }
ρt , t ∈ R ←’ vt , t ∈ R
De¬nition A.2 When vt = v is independent of t, the associated isotopy is called
the exponential map or the ¬‚ow of v and is denoted exp tv; i.e., {exp tv : M ’
M | t ∈ R} is the unique smooth family of di¬eomorphisms satisfying
d
exp tv|t=0 = idM and (exp tv)(p) = v(exp tv(p)) .
dt

91
92 APPENDIX A. PREREQUISITES FROM DIFFERENTIAL GEOMETRY

De¬nition A.3 The Lie derivative is the operator
d
(exp tv)— ω|t=0 .
Lv : „¦k (M ) ’’ „¦k (M ) Lv ω :=
de¬ned by
dt
When a vector ¬eld vt is time-dependent, its ¬‚ow, that is, the corresponding
isotopy ρ, still locally exists by Picard™s theorem. More precisely, in the neigh-
borhood of any point p and for su¬ciently small time t, there is a one-parameter
family of local di¬eomorphisms ρt satisfying
dρt
= vt —¦ ρt and ρ0 = id .
dt
Hence, we say that the Lie derivative by vt is
d
(ρt )— ω|t=0 .
Lvt : „¦k (M ) ’’ „¦k (M ) Lvt ω :=
de¬ned by
dt

Exercise 42
Prove the Cartan magic formula,
Lv ω = ±v dω + d±v ω ,
and the formula
d—
ρt ω = ρ — L v t ω , ()
t
dt
where ρ is the (local) isotopy generated by vt . A good strategy for each formula
is to follow the steps:
(a) Check the formula for 0-forms ω ∈ „¦ 0 (M ) = C ∞ (M ).
(b) Check that both sides commute with d.
(c) Check that both sides are derivations of the algebra („¦— (M ), §). For
instance, check that
Lv (ω § ±) = (Lv ω) § ± + ω § (Lv ±) .

(d) Notice that, if U is the domain of a coordinate system, then „¦ • (U ) is
generated as an algebra by „¦0 (U ) and d„¦0 (U ), i.e., every element in
„¦• (U ) is a linear combination of wedge products of elements in „¦ 0 (U )
and elements in d„¦0 (U ).


We will need the following improved version of formula ( ).

Proposition A.4 For a smooth family ωt , t ∈ R, of d-forms, we have
d— dωt
ρ t ωt = ρ — L v t ωt + .
t
dt dt

Proof. If f (x, y) is a real function of two variables, by the chain rule we have
d d d
f (t, t) = f (x, t) + f (t, y) .
dt dx dy
x=t y=t
93
A.2. SUBMANIFOLDS

Therefore,
d— d— d—
ρ ωt = ρ ωt + ρ ωy
dt t dx x dy t
x=t y=t

dωy
ρ— L v x ω t by ( ) ρ—
x t dy
x=t y=t



dωt
= ρ — L v t ωt + .
t
dt




A.2 Submanifolds

Let M and X be manifolds with dim X < dim M .

De¬nition A.5 A map i : X ’ M is an immersion if di p : Tp X ’ Ti(p) M is
injective for any point p ∈ X.
An embedding is an immersion which is a homeomorphism onto its image. 1
A closed embedding is a proper2 injective immersion.


Exercise 43
Show that a map i : X ’ M is a closed embedding if and only if i is an
embedding and its image i(X) is closed in M .
Hint:
• If i is injective and proper, then for any neighborhood U of p ∈ X, there
is a neighborhood V of i(p) such that f ’1 (V) ⊆ U .
• On a Hausdor¬ space, any compact set is closed. On any topological
space, a closed subset of a compact set is compact.
• An embedding is proper if and only if its image is closed.




De¬nition A.6 A submanifold of M is a manifold X with a closed embedding
i : X ’ M .3

Notation. Given a submanifold, we regard the embedding i : X ’ M as an
inclusion, in order to identify points and tangent vectors:

Tp X = dip (Tp X) ‚ Tp M .
p = i(p) and
1 Theimage has the topology induced by the target manifold.
2A map is proper if the preimage of any compact set is compact.
3 When X is an open subset of a manifold M , we refer to it as an open submanifold.
94 APPENDIX A. PREREQUISITES FROM DIFFERENTIAL GEOMETRY

A.3 Tubular Neighborhood Theorem
Let M be an n-dimensional manifold, and let X be a k-dimensional submanifold
where k < n and with inclusion map
i:X ’M .
At each x ∈ X, the tangent space to X is viewed as a subspace of the tangent
space to M via the linear inclusion dix : Tx X ’ Tx M , where we denote x = i(x).
The quotient Nx X := Tx M/Tx X is an (n ’ k)-dimensional vector space, known
as the normal space to X at x. The normal bundle of X is
N X = {(x, v) | x ∈ X , v ∈ Nx X} .
The set N X has the structure of a vector bundle over X of rank n ’ k under the
natural projection, hence as a manifold N X is n-dimensional.

Exercises 44
Let M be Rn and let X be a k-dimensional compact submanifold of R n .
(a) Show that in this case Nx X can be identi¬ed with the usual “normal
space” to X in Rn , that is, the orthogonal complement in R n of the
tangent space to X at x.
(b) Given µ > 0 let Uµ be the set of all points in Rn which are at a distance
less than µ from X. Show that, for µ su¬ciently small, every point p ∈ U µ
has a unique nearest point π(p) ∈ X.
(c) Let π : Uµ ’ X be the map de¬ned in the previous exercise for µ
su¬ciently small. Show that, if p ∈ Uµ , then the line segment (1 ’ t) ·
p + t · π(p), 0 ¤ t ¤ 1, joining p to π(p) lies in Uµ .
(d) Let N Xµ = {(x, v) ∈ N X such that |v| < µ}. Let exp : N X ’ Rn be
the map (x, v) ’ x + v, and let ν : N Xµ ’ X be the map (x, v) ’ x.
Show that, for µ su¬ciently small, exp maps N X µ di¬eomorphically
onto Uµ , and show also that the following diagram commutes:

E
exp
N Xµ Uµ

d  
d  
νd  π

d  
©
X
(e) Suppose now that the manifold X is not compact. Prove that the as-
sertion about exp is still true provided we replace µ by a continuous
function
µ : X ’ R+
which tends to zero fast enough as x tends to in¬nity. You have thus
proved the tubular neighborhood theorem in Rn .

In general, the zero section of N X,
i0 : X ’ N X , x ’ (x, 0) ,
95
A.3. TUBULAR NEIGHBORHOOD THEOREM

embeds X as a closed submanifold of N X. A neighborhood U0 of the zero section
X in N X is called convex if the intersection U0 © Nx X with each ¬ber is convex.

Theorem A.7 (Tubular Neighborhood Theorem) Let M be an n-dimensio-
nal manifold, X a k-dimensional submanifold, N X the normal bundle of X in M ,
i0 : X ’ N X the zero section, and i : X ’ M inclusion. Then there exist a
convex neighborhood U0 of X in N X, a neighborhood U of X in M , and a di¬eo-
morphism • : U0 ’ U such that

• E U ⊆M
N X ⊇ U0
d
s  

d  
d   commutes.
i0 d  i
d  
X
Outline of the proof.

• Case of M = Rn , and X is a compact submanifold of Rn .

Theorem A.8 (µ-Neighborhood Theorem)
Let U µ = {p ∈ Rn : |p ’ q| < µ for some q ∈ X} be the set of points at a
distance less than µ from X. Then, for µ su¬ciently small, each p ∈ U µ has
a unique nearest point q ∈ X (i.e., a unique q ∈ X minimizing |q ’ x|).
π
Moreover, setting q = π(p), the map U µ ’ X is a (smooth) submersion with
the property that, for all p ∈ U µ , the line segment (1 ’ t)p + tq, 0 ¤ t ¤ 1, is
in U µ .

Here is a sketch. At any x ∈ X, the normal space Nx X may be regarded as
an (n ’ k)-dimensional subspace of Rn , namely the orthogonal complement
in Rn of the tangent space to X at x:

{v ∈ Rn : v ⊥ w , for all w ∈ Tx X} .
Nx X

We de¬ne the following open neighborhood of X in N X:

N X µ = {(x, v) ∈ N X : |v| < µ} .

Let
’’ Rn
exp : NX
’’ x + v .
(x, v)
Restricted to the zero section, exp is the identity map on X.
96 APPENDIX A. PREREQUISITES FROM DIFFERENTIAL GEOMETRY

Prove that, for µ su¬ciently small, exp maps N X µ di¬eomorphically onto
U µ , and show also that the diagram
exp E Uµ
NXµ
d  
d  
d   commutes.
π0 d  π

d  
©
X
• Case where X is a compact submanifold of an arbitrary manifold M .
Put a riemannian metric g on M , and let d(p, q) be the riemannian distance
between p, q ∈ M . The µ-neighborhood of a compact submanifold X is
U µ = {p ∈ M | d(p, q) < µ for some q ∈ X} .
Prove the µ-neighborhood theorem in this setting: for µ small enough, the
following assertions hold.
“ Any p ∈ U µ has a unique point q ∈ X with minimal d(p, q). Set q = π(p).
π
“ The map U µ ’ X is a submersion and, for all p ∈ U µ , there is a unique
geodesic curve γ joining p to q = π(p).
“ The normal space to X at x ∈ X is naturally identi¬ed with a subspace of
Tx M :
Nx X {v ∈ Tx M | gx (v, w) = 0 , for any w ∈ Tx X} .
Let N X µ = {(x, v) ∈ N X | gx (v, v) < µ}.
“ De¬ne exp : N X µ ’ M by exp(x, v) = γ(1), where γ : [0, 1] ’ M is the
geodesic with γ(0) = x and dγ (0) = v. Then exp maps N X µ di¬eomorphi-
dt
cally to U µ .
• General case.
When X is not compact, adapt the previous argument by replacing µ by an
appropriate continuous function µ : X ’ R+ which tends to zero fast enough
as x tends to in¬nity.


Restricting to the subset U 0 ⊆ N X from the tubular neighborhood theorem,
π0 ’1
we obtain a submersion U0 ’’ X with all ¬bers π0 (x) convex. We can carry this
¬bration to U by setting π = π0 —¦ •’1 :
U0 ⊆ NX U ⊆M
is a ¬bration =’ is a ¬bration
π0 “ π“
X X
This is called the tubular neighborhood ¬bration.
97
A.4. HOMOTOPY FORMULA

A.4 Homotopy Formula

Let U be a tubular neighborhood of a submanifold X in M . The restriction i— :
d d
HdeRham (U) ’ HdeRham (X) by the inclusion map is surjective. As a corollary
of the tubular neighborhood ¬bration, i— is also injective: this follows from the
homotopy-invariance of de Rham cohomology.

Corollary A.9 For any degree , HdeRham (U) HdeRham (X).

At the level of forms, this means that, if ω is a closed -form on U and i— ω is
exact on X, then ω is exact. We will need the following related result.

Proposition A.10 If a closed -form ω on U has restriction i— ω = 0, then ω is
exact, i.e., ω = dµ for some µ ∈ „¦d’1 (U). Moreover, we can choose µ such that
µx = 0 at all x ∈ X.


Proof. Via • : U0 ’’ U, it is equivalent to work over U0 . De¬ne for every 0 ¤ t ¤ 1
a map
U0 ’’ U0
ρt :
(x, v) ’’ (x, tv) .
This is well-de¬ned since U0 is convex. The map ρ1 is the identity, ρ0 = i0 —¦π0 , and
each ρt ¬xes X, that is, ρt —¦ i0 = i0 . We hence say that the family {ρt | 0 ¤ t ¤ 1}
is a homotopy from i0 —¦ π0 to the identity ¬xing X. The map π0 : U0 ’ X is
called a retraction because π0 —¦ i0 is the identity. The submanifold X is then
called a deformation retract of U.
A (de Rham) homotopy operator between ρ0 = i0 —¦ π0 and ρ1 = id is a
linear map
Q : „¦d (U0 ) ’’ „¦d’1 (U0 )
satisfying the homotopy formula

Id ’ (i0 —¦ π0 )— = dQ + Qd .

When dω = 0 and i— ω = 0, the operator Q gives ω = dQω, so that we can take
0
µ = Qω. A concrete operator Q is given by the formula:
1
ρ— (±vt ω) dt ,
Qω = t
0

where vt , at the point q = ρt (p), is the vector tangent to the curve ρs (p) at s = t.
The proof that Q satis¬es the homotopy formula is below.
In our case, for x ∈ X, ρt (x) = x (all t) is the constant curve, so vt vanishes
at all x for all t, hence µx = 0.
98 APPENDIX A. PREREQUISITES FROM DIFFERENTIAL GEOMETRY

To check that Q above satis¬es the homotopy formula, we compute
1 1
ρ— (±vt dω)dt ρ— (±vt ω)dt
Qdω + dQω = +d
t t
0 0

1
ρ— (±vt dω + d±vt ω )dt ,
= t
0
Lvt ω


where Lv denotes the Lie derivative along v (reviewed in the next section), and we
used the Cartan magic formula: Lv ω = ±v dω + d±v ω. The result now follows from
d—
ρ t ω = ρ — Lvt ω
t
dt
and from the fundamental theorem of calculus:
1
d—
ρt ω dt = ρ— ω ’ ρ— ω .
Qdω + dQω = 1 0
dt
0


A.5 Whitney Extension Theorem
Theorem A.11 (Whitney Extension Theorem) Let M be an n-dimensional
manifold and X a k-dimensional submanifold with k < n. Suppose that at each
p ∈ X we are given a linear isomorphism Lp : Tp M ’ Tp M such that Lp |Tp X =
IdTp X and Lp depends smoothly on p. Then there exists an embedding h : N ’ M
of some neighborhood N of X in M such that h|X = idX and dhp = Lp for all
p ∈ X.

The linear maps L serve as “germs” for the embedding.
Sketch of proof for the Whitney theorem.
Case M = Rn : For a compact k-dimensional submanifold X, take a neigh-
borhood of the form

U µ = {p ∈ M | distance (p, X) ¤ µ} .

For µ su¬ciently small so that any p ∈ U µ has a unique nearest point in X, de¬ne a
projection π : U µ ’ X, p ’ point on X closest to p. If π(p) = q, then p = q +v for
some v ∈ Nq X where Nq X = (Tq X)⊥ is the normal space at q; see Appendix A.
Let
h : U µ ’’ Rn
p ’’ q + Lq v ,
where q = π(p) and v = p ’ π(p) ∈ Nq X. Then hX = idX and dhp = Lp for p ∈ X.
If X is not compact, replace µ by a continuous function µ : X ’ R+ which tends
to zero fast enough as x tends to in¬nity.
99
A.5. WHITNEY EXTENSION THEOREM

General case: Choose a riemannian metric on M . Replace distance by rieman-
nian distance, replace straight lines q + tv by geodesics exp(q, v)(t) and replace
q + Lq v by the value at t = 1 of the geodesic with initial value q and initial velocity
Lq v.
Appendix B

Prerequisites from Lie
Group Actions

B.1 One-Parameter Groups of Di¬eomorphisms
Let M be a manifold and X a complete vector ¬eld on M . Let ρt : M ’ M , t ∈ R,
be the family of di¬eomorphisms generated by X. For each p ∈ M , ρt (p), t ∈ R,
is by de¬nition the unique integral curve of X passing through p at time 0, i.e.,
ρt (p) satis¬es
±
 ρ0 (p) = p

 dρt (p)
 = X(ρt (p)) .
dt
Claim. We have that ρt —¦ ρs = ρt+s .

Proof. Let ρs (q) = p. We need to show that (ρt —¦ ρs )(q) = ρt+s (q), for all t ∈ R.
Reparametrize as ρt (q) := ρt+s (q). Then
˜
±
 ρ0 (q) = ρs (q) = p
˜


ρ
 d˜t (q) dρt+s (q)
= = X(ρt+s (q)) = X(˜t (q)) ,
ρ
dt dt
i.e., ρt (q) is an integral curve of X through p. By uniqueness we must have ρt (q) =
˜ ˜
ρt (p), that is, ρt+s (q) = ρt (ρs (q)).

Consequence. We have that ρ’1 = ρ’t .
t

In terms of the group (R, +) and the group (Di¬(M ), —¦) of all di¬eomorphisms
of M , these results can be summarized as:

101
102 APPENDIX B. PREREQUISITES FROM LIE GROUP ACTIONS

Corollary B.1 The map R ’ Di¬(M ), t ’ ρt , is a group homomorphism.

The family {ρt | t ∈ R} is then called a one-parameter group of di¬eo-
morphisms of M and denoted

ρt = exp tX .


B.2 Lie Groups

De¬nition B.2 A Lie group is a manifold G equipped with a group structure
where the group operations

G — G ’’ G G ’’ G
and
a ’’ a’1
(a, b) ’’ a · b

are smooth maps.


Examples.

• R (with addition1 ).

• S 1 regarded as unit complex numbers with multiplication, represents rota-
tions of the plane: S 1 = U(1) = SO(2).

• U(n), unitary linear transformations of Cn .

• SU(n), unitary linear transformations of Cn with det = 1.

• O(n), orthogonal linear transformations of Rn .

• SO(n), elements of O(n) with det = 1.

• GL(V ), invertible linear transformations of a vector space V .

™¦


De¬nition B.3 A representation of a Lie group G on a vector space V is a
group homomorphism G ’ GL(V ).
1 The operation will be omitted when it is clear from the context.
103
B.3. SMOOTH ACTIONS

B.3 Smooth Actions
Let M be a manifold.
De¬nition B.4 An action of a Lie group G on M is a group homomorphism
ψ : G ’’ Di¬(M )
g ’’ ψg .
(We will only consider left actions where ψ is a homomorphism. A right action
is de¬ned with ψ being an anti-homomorphism.) The evaluation map associated
with an action ψ : G ’ Di¬(M ) is
evψ : M — G ’’ M
(p, g) ’’ ψg (p) .
The action ψ is smooth if ev ψ is a smooth map.

Example. If X is a complete vector ¬eld on M , then
ρ : R ’’ Di¬(M )
t ’’ ρt = exp tX
™¦
is a smooth action of R on M .
Every complete vector ¬eld gives rise to a smooth action of R on M . Con-
versely, every smooth action of R on M is de¬ned by a complete vector ¬eld.
{complete vector ¬elds on M} ←’ {smooth actions of R on M }

’’
X exp tX

dψt (p)
←’
Xp = ψ
dt t=0


B.4 Adjoint and Coadjoint Representations
Let G be a Lie group. Given g ∈ G let
Lg : G ’’ G
a ’’ g · a
be left multiplication by g. A vector ¬eld X on G is called left-invariant if
(Lg )— X = X for every g ∈ G. (There are similar right notions.)
Let g be the vector space of all left-invariant vector ¬elds on G. Together
with the Lie bracket [·, ·] of vector ¬elds, g forms a Lie algebra, called the Lie
algebra of the Lie group G.
104 APPENDIX B. PREREQUISITES FROM LIE GROUP ACTIONS

Exercise 45
Show that the map
’’ Te G
g
X ’’ Xe
where e is the identity element in G, is an isomorphism of vector spaces.

Any Lie group G acts on itself by conjugation:
G ’’ Di¬(G)
ψg (a) = g · a · g ’1 .
g ’’ ψg ,
The derivative at the identity of
ψg : G ’’ G
a ’’ g · a · g ’1
is an invertible linear map Adg : g ’’ g. Here we identify the Lie algebra g with
the tangent space Te G. Letting g vary, we obtain the adjoint representation (or
adjoint action) of G on g:
Ad : G ’’ GL(g)
g ’’ Adg .

Exercise 46
Check for matrix groups that
d
Adexp tX Y = [X, Y ] , ∀X, Y ∈ g .
dt t=0

Hint: For a matrix group G (i.e., a subgroup of GL(n; R) for some n), we have
Adg (Y ) = gY g ’1 , ∀g ∈ G , ∀Y ∈ g
and
[X, Y ] = XY ’ Y X , ∀X, Y ∈ g .

Let ·, · be the natural pairing between g— and g:
·, · : g— — g ’’ R
(ξ, X) ’’ ξ, X = ξ(X) .
Given ξ ∈ g— , we de¬ne Ad— ξ by
g

Ad— ξ, X = ξ, Adg’1 X , for any X ∈ g .
g

The collection of maps Ad— forms the coadjoint representation (or coadjoint
g
action) of G on g— :
Ad— : G ’’ GL(g— )
g ’’ Ad— . g

We take g ’1 in the de¬nition of Ad— ξ in order to obtain a (left) representation,
g
i.e., a group homomorphism, instead of a “right” representation, i.e., a group anti-
homomorphism.
105
B.5. ORBIT SPACES

Exercise 47
Ad— —¦ Ad— = Ad— .
Show that Adg —¦ Adh = Adgh and g h gh




B.5 Orbit Spaces
Let ψ : G ’ Di¬(M ) be any action.

De¬nition B.5 The orbit of G through p ∈ M is {ψg (p) | g ∈ G}. The stabi-
lizer (or isotropy) of p ∈ M is the subgroup Gp := {g ∈ G | ψg (p) = p}.

Exercise 48
If q is in the orbit of p, then Gq and Gp are conjugate subgroups.



De¬nition B.6 We say that the action of G on M is . . .

• transitive if there is just one orbit,

• free if all stabilizers are trivial {e},
• locally free if all stabilizers are discrete.

Let ∼ be the orbit equivalence relation; for p, q ∈ M ,

p∼q ⇐’ p and q are on the same orbit.

The space of orbits M/ ∼ = M/G is called the orbit space. Let

π : M ’’ M/G
p ’’ orbit through p

be the point-orbit projection.
Topology of the orbit space:
We equip M/G with the weakest topology for which π is continuous, i.e.,
U ⊆ M/G is open if and only if π ’1 (U) is open in M . This is called the quotient
topology. This topology can be “bad.” For instance:
Example. Let G = R act on M = R by

t ’’ ψt = multiplication by et .

There are three orbits R+ , R’ and {0}. The point in the three-point orbit space
corresponding to the orbit {0} is not open, so the orbit space with the quotient
™¦
topology is not Hausdor¬.
106 APPENDIX B. PREREQUISITES FROM LIE GROUP ACTIONS

n
Example. Let G = C \{0} act on M = C by

» ’’ ψ» = multiplication by » .

The orbits are the punctured complex lines (through non-zero vectors z ∈ Cn ),
plus one “unstable” orbit through 0, which has a single point. The orbit space is
n’1
{point} .
M/G = C P
n’1
The quotient topology restricts to the usual topology on C P . The only open set
containing {point} in the quotient topology is the full space. Again the quotient
topology in M/G is not Hausdor¬.
However, it su¬ces to remove 0 from Cn to obtain a Hausdor¬ orbit space:
n’1
C P . Then there is also a compact (yet not complex) description of the orbit
space by taking only unit vectors:

C \{0} = S 2n’1 /S 1 .
n’1
= Cn \{0}
CP

™¦
Appendix C

Variational Principles

C.1 Principle of Least Action

The equations of motion in classical mechanics arise as solutions of variational
problems. For a general mechanical system of n particles in R3 , the physical path
satis¬es Newton™s second law. On the other hand, the physical path minimizes the
mean value of kinetic minus potential energy. This quantity is called the action.
For a system with constraints, the physical path is the path which minimizes the
action among all paths satisfying the constraint.
Example. Suppose that a point-particle of mass m moves in R3 under a force
¬eld F ; let x(t), a ¤ t ¤ b, be its path of motion in R3 . Newton™s second law states
that
d2 x
m 2 (t) = F (x(t)) .
dt
De¬ne the work of a path γ : [a, b] ’’ R3 , with γ(a) = p and γ(b) = q, to be
b

F (γ(t)) ·
Wγ = (t)dt .
dt
a

Suppose that F is conservative, i.e., Wγ depends only on p and q. Then we can
de¬ne the potential energy V : R3 ’’ R of the system as

V (q) := Wγ

where γ is a path joining a ¬xed base point p0 ∈ R3 (the “origin”) to q. Newton™s
second law can now be written
d2 x ‚V
m 2 (t) = ’ (x(t)) .
dt ‚x

107
108 APPENDIX C. VARIATIONAL PRINCIPLES

In Lecture 4 we saw that
⇐’
Newton™s second law Hamilton equations
in R3 = {(q1 , q2 , q3 )} in T — R3 = {(q1 , q2 , q3 , p1 , p2 , p3 )}

where pi = m dqi and the hamiltonian is H(p, q) = 2m |p|2 + V (q). Hence, solving
1
dt
Newton™s second law in con¬guration space R3 is equivalent to solving in phase
space for the integral curve T — R3 of the hamiltonian vector ¬eld with hamiltonian
™¦
function H.

Example. The motion of earth about the sun, both regarded as point-masses and
assuming that the sun to be stationary at the origin, obeys the inverse square
law
d2 x ‚V
m 2 =’ ,
dt ‚x
where x(t) is the position of earth at time t, and V (x) = const. is the gravita-
|x|
™¦
tional potential.

When we need to deal with systems with constraints, such as the simple pen-
dulum, or two point masses attached by a rigid rod, or a rigid body, the language
of variational principles becomes more appropriate than the explicit analogues
of Newton™s second laws. Variational principles are due mostly to D™Alembert,
Maupertius, Euler and Lagrange.
Example. (The n-particle system.) Suppose that we have n point-particles
of masses m1 , . . . , mn moving in 3-space. At any time t, the con¬guration of this
system is described by a vector in con¬guration space R3n
x = (x1 , . . . , xn ) ∈ R3n
with xi ∈ R3 describing the position of the ith particle. If V ∈ C ∞ (R3n ) is the
potential energy, then a path of motion x(t), a ¤ t ¤ b, satis¬es
d 2 xi ‚V
(t) = ’
mi (x1 (t), . . . , xn (t)) .
dt2 ‚xi
Consider this path in con¬guration space as a map γ0 : [a, b] ’ R3n with γ0 (a) = p
and γ0 (b) = q, and let
P = {γ : [a, b] ’’ R3n | γ(a) = p and γ(b) = q}
be the set of all paths going from p to q over time t ∈ [a, b]. ™¦

De¬nition C.1 The action of a path γ ∈ P is
2
b
mi dγi
Aγ := ’ V (γ(t)) dt .
(t)
2 dt
a
109
C.2. VARIATIONAL PROBLEMS

Principle of least action.
The physical path γ0 is the path for which Aγ is minimal.
Newton™s second law for a constrained system.
Suppose that the n point-masses are restricted to move on a submanifold
M of R3n called the constraint set. We can now single out the actual physical
path γ0 : [a, b] ’ M , with γ0 (a) = p and γ0 (b) = q, as being “the” path which
minimizes Aγ among all those hypothetical paths γ : [a, b] ’ R3n with γ(a) = p,
γ(b) = q and satisfying the rigid constraints γ(t) ∈ M for all t.


C.2 Variational Problems
Let M be an n-dimensional manifold. Its tangent bundle T M is a 2n-dimensional
manifold. Let F : T M ’ R be a smooth function.
If γ : [a, b] ’ M is a smooth curve on M , de¬ne the lift of γ to T M to be
the smooth curve on T M given by

γ : [a, b] ’’ T M
˜
γ(t), dγ (t)
t ’’ .
dt


The action of γ is
b b


Aγ := (˜ F )(t)dt =
γ F γ(t), (t) dt .
dt
a a

For ¬xed p, q ∈ M , let

P(a, b, p, q) := {γ : [a, b] ’’ M | γ(a) = p, γ(b) = q} .

Problem.
Find, among all γ ∈ P(a, b, p, q), the curve γ0 which “minimizes” Aγ .
First observe that minimizing curves are always locally minimizing:

Lemma C.2 Suppose that γ 0 : [a, b] ’ M is minimizing. Let [a1 , b1 ] be a subin-
terval of [a, b] and let p1 = γ0 (a1 ), q1 = γ0 (b1 ). Then γ0 |[a1 ,b1 ] is minimizing among
the curves in P(a1 , b1 , p1 , q1 ).

Proof. Exercise:
Argue by contradiction. Suppose that there were γ1 ∈ P(a1 , b1 , p1 , q1 ) for
which Aγ1 < Aγ0 |[a1 ,b1 ] . Consider a broken path obtained from γ0 by replacing
the segment γ0 |[a1 ,b1 ] by γ1 . Construct a smooth curve γ2 ∈ P(a, b, p, q) for which
Aγ2 < Aγ0 by rounding o¬ the corners of the broken path.
110 APPENDIX C. VARIATIONAL PRINCIPLES

We now assume that p, q and γ0 lie in a coordinate neighborhood (U, x1 , . . . , xn ).
On T U we have coordinates (x1 , . . . , xn , v1 , . . . , vn ) associated with a trivialization
‚ ‚
of T U by ‚x1 , . . . , ‚xn . Using this trivialization, the curve

γ : [a, b] ’’ U , γ(t) = (γ1 (t), . . . , γn (t))

lifts to
dγ1 dγn
γ : [a, b] ’’ T U ,
˜ γ (t) =
˜ γ1 (t), . . . , γn (t), (t), . . . , (t) .
dt dt

Necessary condition for γ0 ∈ P(a, b, p, q) to minimize the action.
Let c1 , . . . , cn ∈ C ∞ ([a, b]) be such that ci (a) = ci (b) = 0. Let γµ : [a, b] ’’ U
be the curve
γµ (t) = (γ1 (t) + µc1 (t), . . . , γn (t) + µcn (t)) .
For µ small, γµ is well-de¬ned and in P(a, b, p, q).
b
Let Aµ = Aγµ = a F γµ (t), dγµ (t) dt. If γ0 minimizes A, then
dt

dAµ
(0) = 0 .

b
dAµ ‚F dγ0 ‚F dγ0 dci
(0) = γ0 (t), (t) ci (t) + γ0 (t), (t) (t) dt
dµ ‚xi dt ‚vi dt dt
a i
b
‚F d ‚F
(. . .) ’
= (. . .) ci (t)dt = 0
‚xi dt ‚vi
a i

where the ¬rst equality follows from the Leibniz rule and the second equality fol-
lows from integration by parts. Since this is true for all ci ™s satisfying the boundary
conditions ci (a) = ci (b) = 0, we conclude that
‚F dγ0 d ‚F dγ0
γ0 (t), (t) = γ0 (t), (t) . (E-L)
‚xi dt dt ‚vi dt
These are the Euler-Lagrange equations.
Example. Let (M, g) be a riemannian manifold. From the riemannian metric, we
get a function F : T M ’ R, whose restriction to each tangent space Tp M is the
quadratic form de¬ned by the metric. On a coordinate chart (U, x1 , . . . , xn ) on M ,
we have
gij (x)v i v j .
F (x, v) =
Let p and q be points on M , and let γ : [a, b] ’ M be a smooth curve joining
p to q. Let γ : [a, b] ’ T M , γ (t) = (γ(t), dγ (t)) be the lift of γ to T M . The action
˜ ˜ dt
of γ is
2
b b


A(γ) = (˜ F ) dt =
γ dt .
dt
a a
111
C.3. SOLVING THE EULER-LAGRANGE EQUATIONS

It is not hard to show that the Euler-Lagrange equations associated to the action
reduce to the Christo¬el equations for a geodesic

d2 γ k dγ i dγ j
(“k —¦ γ)
+ =0,
ij
dt2 dt dt

where the “k ™s (called the Christo¬el symbols) are de¬ned in terms of the
ij
coe¬cients of the riemannian metric by

1 ‚g i ‚g j ‚gij
“k = k

g + ,
ij
2 ‚xj ‚xi ‚x

(g ij ) being the matrix inverse to (gij ). ™¦



C.3 Solving the Euler-Lagrange Equations

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