1. Jacobi fields
Definition 1.1. Let c : [0, 1] → M be a curve in M . A Jacobi field along c is a
D2
vector field J satisfying dt
2 J + R(J, ċ)ċ = 0.
Jacobi fields are important because they characterize families of geodesics:
Proposition 1.2. Let s : [0, 1] × (−ε, ε) be a surface, so that t 7→ s(t, 0) is a
∂s
geodesic. Then t 7→ s(t, u) is a geodesic for all u ∈ (−ε, ε) if and only if ∂u
is a
Jacobi field everywhere.
Proof: For any surface s,
D D ∂s
D D ∂s
∂s ∂s ∂s
D D ∂s
∂s ∂s ∂s
=
+R
,
=
+R
,
.
du dt ∂t
dt du ∂t
∂u ∂t ∂t
dt dt ∂u
∂u ∂t ∂t
D D ∂s
∂s
Therefore, du
dt ∂t = 0 if and only if ∂u is a Jacobi field. By definition t 7→ s(t, u)
D ∂s
is a geodesic if and only if dt ∂t = 0. Since this is assumed to be true at u = 0,
D ∂s
we see that it is true everywhere if and only if dt
∂t is constant with respect to u,
∂s
which happens if and only if ∂u is Jacobi.
On a complete manifold it’s also true that any Jacobi field along a geodesic
arises this way, as a 1-parameter family of geodesics. First we show this in a
small neighborhood. The space of Jacobi vector fields along a curve is always a
2n-dimensional vector space, since it is the space of solutions to a second order
ordinary differential equation in n variables. In a small neighborhood, there is a
unique geodesic connecting any two points. Therefore, along a geodesic connecting
p to q, the restriction map {Jacobi vector fields} → T Mp ⊕ T Mq is surjective: any
1-parameter family of endpoints is realized by a 1-parameter family of geodesics.
But since the dimensions are equal the restriction map is an isomorphism.
The local statement implies the global statement. Given a Jacobi field J along
a (not necessarily small) geodesic ω : [0, 1] → M , the restriction of J to [0, δ] is
a Jacobi field along the restricted geodesic. By the above, we see that J|[ 0, δ] is
realized by a family of geodesics. By completeness we can extend these geodesics
to [0, 1]. This family of geodesics defines a Jacobi vector field Je along all of ω, but
since Je = J on [0, δ] we conclude that Je = J everywhere (J is determined by its
initial conditions.
Example 1.3. Note however that it is not generally true that geodesics are determined by their endpoints, or that {Jacobi vector fields} → T Mp ⊕ T Mq is surjective. For example, let M be the round 2-sphere of radius 1 in R3 , and let
p and q be the north and south poles. We can parametrize S 2 by (x, y, z) =
(cos θ cos ϕ, sin θ cos ϕ, sin ϕ). It’s easy to see that h∂θ , ∂ϕ i = 0, also ||∂θ ||2 = cos2 ϕ
and ||∂ϕ ||2 = 1. It follows that ∇∂θ ∂ϕ = ∇∂ϕ ∂θ = − tan ϕ∂θ , ∇∂ϕ ∂ϕ = 0, and
∇∂θ ∂θ = cos ϕ sin ϕ∂ϕ .
Therefore R(∂θ , ∂ϕ )∂ϕ = ∇∂ϕ tan ϕ∂θ = sec2 ϕ∂θ −tan2 ϕ∂θ = ∂θ . By symmetries
this completely determines R. Notice that for constant θ the curve ϕ 7→ (θ, ϕ) is a
geodesic, since ∇∂ϕ ∂ϕ = 0. We also calculate ∇∂ϕ ∇∂ϕ ∂θ = −∂θ , seeing explicitly
that ∂θ is a Jacobi vector field for these curves.
2. The path space and the energy functional
Let M be a manifold, and fix points p, q ∈ M . Let Ω(p, q) denote the space of
curves in M connecting p to q. Of course there are many different spaces of paths
1
2
and topologies we can look at: continuous paths, C k paths, etc. However, we are
mostly interested in the homotopy type of this space, and for these purposes the
choice does not matter: it turns out that any reasonable definition of “path space”
will have the same homotopy type.
For our purposes, we will always look at the space of piecewise C ∞ paths, with
domain [0, 1]. In particular, if ω ∈ Ω(p, q), the velocity ω̇ is defined everywhere
outside of a discrete set. To define a topology on this space, we use a type of C 1
metric: if ω0 and ω1 are paths, define
Z
ρ(ω0 , ω1 ) = max ρ(ω0 (t), ω1 (t)) +
t∈[0,1]
1
21
(||ω̇0 || − ||ω̇1 ||) dt
2
0
.
However, we will almost never work with this metric directly. To understand
the topology of Ω(p, q) (which we will often denote simply by Ω), we work with the
following function.
Definition 2.1. Let ω ∈ Ω and let 0 6 a < b 6 1. We say the energy of ω from a
to b is
Z b
Eab (ω) =
||ω̇||2 dt.
a
When a = 0 and b = 1, we will simply say the energy of ω without referring to the
endpoints, and denote this by E(ω).
Notice that Eab : Ω → R is a continuous function for any a, b. To understand
the topology of Ω, we will think of E as a Morse function on Ω, and use the main
theorems of Morse theory to compute its homotopy type.
Of course, some explanation is needed here, because Ω is certainly not a smooth
manifold. Indeed it is not even locally compact. But in some sense this is the “only
reason” Ω fails to be a manifold, and we often think of it as an “infinite dimensional
manifold”.
With this in mind, we should define T Ωω ), the tangent space to Ω at ω. A
tangent vector at a point is just the derivative of a path passing through that
point. A path in Ω is a path of paths in M , that is, a surface in M . With this in
mind, we make the following definition.
Definition 2.2. If ω ∈ Ω(p, q) a variation of ω is a continuous function α : [0, 1] ×
(−ε, ε) → M , so that α(t, 0) = ω(t), α(0, u) = p, α(1, u) = q. We require that there
is a subdivision 0 = t0 < t1 < . . . < tk = 1 so that α is C ∞ on each subdomain
[tj , tj+1 ] × (−ε, ε).
For fixed u ∈ (−ε, ε), we see that the curve t 7→ α(t, u) is an element of Ω(p, q).
We can therefore also think of α as defining a map α : (−ε, ε) → Ω. (Milnor uses α
to distinguish α : [0, 1] × (−ε, ε) → M from α : (−ε, ε) → Ω, here we use the same
notation for both.)
If α is a variation at ω ∈ Ω, then ∂α
∂u u=0 is a piecewise smooth vector field along
ω, which vanishes at the endpoints. This motivates the next definition.
Definition 2.3. Let ω ∈ Ω. We define the tangent space of Ω at ω to be the
(infinite dimensional) vector space of all piecewise smooth vector fields along ω
which vanish at the endpoints. We denote this space by T Ωω .
3
The point of this lengthy set-up is so that we can calculate the derivative of
E, dE : T Ω → T R. Let us introduce the following notation. For ω ∈ Ω, let
∆ω̇(t) = ω̇(t+ ) − ω̇(t− ), the discontinuity in the velocity vector at t ∈ [0, 1] (if ω is
smooth at t then ∆ω̇(t) = 0). In general, we will always use dots for derivative with
respect to t ∈ [0, 1], the time parameter of paths in M , and primes for
derivatives
with respect to u. In particular, if α is a variation we let α0 (0) = ∂α
∂u u=0 ∈ T Ωω .
Proposition 2.4. Let α be a variation of the path ω ∈ Ω, then E ◦ α : (−ε, ε) → R.
The derivative of this function is
!
Z 1
X
dE ◦ α
0
0
hα (t, 0), ω̈(t)idt .
= −2
hα (t, 0), ∆ω̇(t)i +
du
0
t
In
the following definition: dEω (W ) =
light of this proposition, we will make
R1
P
−2
hW (t), ∆ω̇(t)i + 0 hW (t), ω̈(t)idt for W ∈ T Ωω . Though this is technit
cally a definition, the proposition says that this is the only definition which makes
sense, if we want the chain rule to hold.
Proof: First we see
Z 1
Z 1
Z 1
dE ◦ α
d
D
D 0
=
hα̇, α̇i dt = 2
α̇, α̇ dt = 2
α , α̇ dt.
du
du 0
du
dt
0
0
D 0 0 D ∂
0
Then using ∂t hα , α̇i = dt α , α̇ + α , dt α̇ , we can use integration by parts to
get
Z tj+1 Z tj+1 D 0
t=t−
j+1
0
0 D
α , α̇ dt = hα , α̇i|t=t+ −
α , α̇ dt.
dt
dt
j
tj
tj
From here we simply sum the equations over j = 0, . . . , k, and set u = 0.
Corollary 2.5. A path ω ∈ Ω is a critical point of E if and only if it is a smooth
geodesic.
Proof: If there is any point where ∆ω̇(t) 6= 0 or ω̈ 6= 0, we can clearly choose
W ∈ T Ωω so that dEω (W ) 6= 0.
3. The Hessian of the energy
Now that we understand the critical set of the energy functional, we would like
to calculate its Hessian on the critical set. This is for two reasons: firstly we want
to see when the Hessian is non-degenerate (so the energy function is Morse), and
secondly we want to be able to calculate the index of the critical points, so that we
understand the homotopy type of the space.
Theorem 3.1. Let α : B 2 (ε) → Ω(p, q) be a 2-parameter family of paths, so that
∂ 2 E◦α α(0) is a geodesic. Then ∂u
=
1 ∂u2
u=0
Z 1
!
X ∂α
D ∂α
∂α D2 ∂α
∂α
,∆
+
,
+R
, α̇ α̇ dt .
−2
∂u2
dt ∂u1
∂u2 dt2 ∂u1
∂u1
0
t
Proof: Starting from Proposition 2.4 we have
Z 1
!
X ∂α
∂2E ◦ α
∂
∂α
= −2
, ∆α̇ +
, α̈ dt
∂u1 ∂u2
∂u1
∂u2
∂u2
0
t
4
X D ∂α
∂α D
= −2
, ∆α̇ +
,
∆α̇
du1 ∂u2
∂u2 du1
t
Z 1
D ∂α
∂α D
+
, α̈ +
,
α̈ dt .
du1 ∂u2
∂u2 du1
0
Since α is assumed to be geodesic at u = 0, we get
Z 1
!
X ∂α D
∂ 2 E ◦ α ∂α D
= −2
,
∆α̇ +
,
α̈ dt .
∂u1 ∂u2 u=0
∂u2 du1
∂u2 du1
0
t
The first term
∂α D
,
∆α̇
∂u2 du1
=
∂α
D ∂α
,∆
∂u2
dt ∂u1
while the second term
∂α D D
∂α D
∂α D2
∂α
,
α̇ =
,
α̈ =
,
α̇ + R
, α̇ α̇
∂u2 du1 dt
∂u2 du1
∂u2 dtdu1
∂u1
∂α D2 ∂α
∂α
=
, 2
+R
, α̇ α̇ .
∂u2 dt ∂u1
∂u1
This completes the proof.
We can rephrase the theorem as saying, at a geodesic ω ∈ Ω and W1 , W2 ∈ T Ωω ,
we have
!
Z 1
X
D
D2
E
H (W1 , W2 ) = −2
W2 , ∆ W1 +
W2 , 2 W1 + R (W1 , ω̇) ω̇ dt .
dt
dt
0
t
Just as in the case of dE, this is technically a definition of H E . But it is the only
definition that is compatible with the chain rule, as the theorem shows.
Corollary 3.2. Let ω ∈ Ω(p, q) be a critical point of E. Then H E is degenerate at
ω if and only if there is a non-trivial Jacobi vector field along ω which vanishes at
p and q. In particular, there is a continuous family of geodesics connecting p to q.
Proof: If W1 is a smooth Jacobi field which vanishes at the endpoints then clearly
H E (W1 , W2 ) = 0 for all W2 ∈ T Ωω . Conversely, if there is any point where
D2
D
E
dt2 W1 +R (W1 , ω̇) ω̇ 6= 0 or ∆ dt W1 6= 0, we can choose W2 so that H (W1 , W2 ) 6= 0.
For the second statement, notice that if any geodesic ω admits a non-trivial
Jacobi vector field which vanishes at the endpoints, then the corresponding surface
∂s
= J will be made of geodesic curves t 7→ s(t, u).
s : [0, 1]×(−ε, ε) → M satisfying ∂u
But s(0, u) = p and s(1, u) = q for all u, since J vanishes there.
Definition 3.3. Let ω be a geodesic connecting p and q. We say that p and q
are conjugate along ω if there is a non-zero Jacobi field along ω which vanishes at
the endpoints. Such Jacobi fields form a vector space, the dimension of this vector
space is called the multiplicity of the conjugacy.
The multiplicity of any conjugacy is at most n − 1 (where n = dim M ). We know
that the space of all Jacobi vector fields is a 2n dimensional space, given by the
initial conditions J(0), DJ
dt (0) ∈ T Mp . Therefore the space of Jacobi vector fields
vanishing at p is n-dimensional. Since J = tω̇ is a Jacobi vector field vanishing at
p but not at q, we see that the dimension of vector fields vanishing at both is at
most n − 1.
A restatement of the above corollary
5
Corollary 3.4. Let ω ∈ Ω(p, q) be a critical point of E. Then H E is degenerate at
ω if and only if p and q are conjugate along ω. In this case, the dimension of the
kernel of H E is exactly the multiplicity of the conjugacy, which is at most n − 1.
4. The index of H E
Theorem 4.1. Let ω ∈ Ω(p, q) be a critical point of E. Then the index of H E is
equal to the total number of points t ∈ [0, 1] so that p and ω(t) are conjugate along
ω|[0,t] , counted with multiplicity. This index is always finite.
Lemma 4.2. Let {Uj } be a covering of ω([0, 1]) by open subsets, with the property
that any two points in Uj are connected by a unique minimal geodesic. Let 0 = t0 <
t1 < . . . < tm = 1 be a partition of [0, 1] so that ω([tj−1 , tj ]) ⊆ Uj . Let T0 ⊆ T Ωω
be the space of all piecewise smooth geodesics which are smooth and Jacobi on each
[tj−1 , tj ], and let T∞ ⊆ T Ωω be the space of all vector fields which vanish at each
tj . Then T Ωω = T0 ⊕ T∞ , T0 is finite dimensional, and H E is positive definite on
T∞ .
Proof: Because in a set Uj geodesics are uniquely determined by their endpoints, we
see that Jacobi vector fields are uniquely determined by their boundary conditions.
Therefore the evaluation map T0 → T Mω(t1 ) ⊕ T Mω(t2 ) ⊕ . . . ⊕ T Mω(tm−1 ) is an
isomorphism. Therefore T0 is finite dimensional. This also implies that T0 ∩ T∞ =
{0}.
Using the inverse of the above isomorphism, if we are given any W ∈ T Ωω , we
can find J ∈ T0 so that J(tj ) = W (tj ) for each j. Then W = J + (W − J) shows
that W ∈ T0 ⊕ T∞ .
It remainsPto show that H E is positive definite on T∞ . Given W ∈ T∞ , we can
m
write W = j=1 Wj , where Wj is non-vanishing only on [tj−1 , tj ] (let Wj = W
on [tj−1 , tj ], and zero elsewhere). From
3.1 we see that H E (Wj , Wi ) = 0
P Theorem
E
E
if j 6= i, which implies H (W, W ) = j H (Wj , Wj ). But ω|[tj−1 ,tj ] is a minimal
geodesic, meaning that it is a local minimum for E, and therefore H E is positive
definite on ω|[tj−1 ,tj ] .
Proof of Theorem 4.1: ω|[0,t] is a geodesic for any t, with this is mind we define
λ(t) to be the index of H E on T Ωω|[0,t] . We prove a series of claims:
• λ : [0, t] → N is an increasing function, and λ(t) = 0 for small t.
• For small ε > 0, λ(t − ε) = λ(t).
• Let k be the dimension of the kernel of H E on T Ωω|[0,t] . Then λ(t) + k =
λ(t + ε) for small ε > 0.
Together these claims imply the theorem.
Suppose t < t0 . Any subspace T ⊆ T Ωω|[0,t] where H E is negative definite defines
a subspace T 0 ⊆ T Ωω|[0,t0 ] where H E is negative definite, simply by extending vector
fields to be zero of ω|[t,t0 ] . This shows that λ is increasing. Since ω|[0,t] is a minimal
geodesic for small t, we see that λ is initially 0, proving the first claim.
This shows that λ(t) > λ(t − ε), so to prove the second claim we only need to
show λ(t) 6 λ(t − ε). According to Lemma 4.2 the index of H E is the same as the
index of H E restricted toT0 , for any partition satisfying the lemma’s hypothesis.
We choose a partition so that t 6= tj for any j, therefore say ti < t < ti+1 .
For any t ∈ (ti , ti+1 ), the evaluation map T0 ∩T Ωω|[0,t] → T Mω(t1 ) ⊕. . .⊕T Mω(ti )
is an isomorphism, whose inverse is given by solving the boundary value problem
6
to the Jacobi equation. To calculate the index of H E restricted to T0 ∩ T Ωω|[0,t] , we
instead think of H E as a time dependent family of symmetric forms on the fixed
vector space T Mti ⊕ . . . ⊕ T Mti . H E is not constant, because the isomorphism
depends on t: solving a boundary value problem for boundary condition 0 at T Mω(t)
is different than solving the same boundary value problem with boundary condition
0 at T Mω(t−ε) . It is however smooth with respect to t; a smooth family of endpoints
gives rise to a smooth family of geodesics, which therefore have smooth derivatives
(the derivative of a family of geodesics are the Jacobi vector fields). But being
negative definite on a subspace is an open condition of matrices: if a matrix A
is negative definite on a subspace then any nearby matrix will also be negative
definite on that subspace. Therefore if H E is negative definite on a λ(t)-dimensional
subspace at t, it will be negative definite on the same subspace at t − ε. This shows
λ(t) 6 λ(t − ε).
Finally we need to show that λ(t) + k = λ(t + ε), where k is the dimension of
the kernel of H E at t. At time t H E is positive definite on a subspace of dimension
dim T0 − λ(t) − k, and since being positive definite is an open condition we know
that H E is also positive definite on the same subspace at time t + ε. This shows
λ(t + ε) 6 λ(t) + k.
At time t, by assumption we have linearly independent vector fields W1 , . . . , Wλ(t)
so that H E is negative definite on the span, and also linearly independent Jacobi
vector fields J1 , . . . Jk which vanish at the endpoints. Since they are determined by
D
their initial conditions at any point, we know that dt
Jj (t) ∈ T Mω(t) are linearly
independent vectors. We then choose k vector fields X1 , . . . , Xk ∈ T Ωω|[0,t+ε] so
DJ
that h dtj (t), Xi i is 1 if i = j and 0 if i 6= j (this is of course possible since it is only
a condition at the one point ω(t)). We extend Wi and Jj to T Ωω|[0,t+ε] by letting
them be zero on [t, t + ε].
Consider the vector fields W1 , . . . , Wλ(t) , δ −1 J1 − δX1 , . . . , δ −1 Jk − δXk . We
claim that for sufficiently small δ > 0, H E is negative definite on their span, which
will complete the proof. With respect to this basis, H E is given by the matrix
H
δA
δA∗ −4 Id +δ 2 B
where Hij = H E (Wi , Wj ), Aij = H E (Wi , Xj ), and Bij = H E (Xi , Xj ). Here we
used the fact that H E (Wi , Jj ) = 0 and H E (Ji , Xj ) = 2 Idij . Since H is assumed
to be a negative definite matrix, it is clear this matrix is negative definite for small
δ.