&%

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

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

dγ

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

dγ

—

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 .

dµ

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

dγ

—

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