SPREADING OF GEODESICS
In what follows (M, g) is a Riemannian manifold of dimension n.
1. Review
Let γ : [0, 1] → M be a curve. We say that V is a vector field along γ if for every
t ∈ [0, 1], V (t) ∈ Tγ(t) M .
Example: If γ is given in local coordinates by (x1 (t), . . . , xn (t)),
V (t) = γ̇(t) =
is a vector field along γ.
n
!
i=1
ẋi
∂
|γ(t)
∂xi
D
We call dt
the covariant derivative along γ associated to the metric g. Recall
that it acts on vector fields along γ, and that DV
dt is a vector field along γ. Also,
D
satisfies
dt
"
# "
#
d
DV
DW
#V, W $ =
, W + V,
dt
dt
dt
for any V and W vector fields along γ.
D
We say that γ is a geodesic if dt
(γ̇) = 0. If γ is the geodesic such that γ(0) = p
and γ̇(0) = v the exponential map expp : Tp M → M satisfies expp (tv) = γ(t).
A parametrized surface in M is a differentiable map f : A ⊂ R2 → M . A
vector field V along the surface f is a map that to every (s, t) assigns a vector
V (s, t) ∈ Tf (s,t) M . Since for every t fixed, s &→ f (s, t) is a curve on M , we define
DV
DV
∂s (s, t) as the covariant derivative of V along the curve s &→ f (s, t). ∂t (s, t) is
defined in the same fashion.
Observe that
$ %
$ %
D df
D df
=
.
∂s ∂t
∂t ∂s
If v, w ∈ Tp M form the basis of a plane σ the sectional curvature at p of σ is
defined by
#R(v, w)v, w$
Kp (σ) = &
3
'v'2 'w'2 − #v, w$
where R is the curvature tensor defined by
R(X, Y )Z = ∇Y ∇X Z − ∇X ∇Y Z + ∇[X,Y ] Z.
In particular, if {v, w} is an orthonormal basis of σ, Kp (σ) = #R(v, w)v, w$.
We remind that R satisfies
$
%
D D
D D
∂f ∂f
V −
V =R
,
V.
∂t ∂s
∂s ∂t
∂s ∂t
1
2
SPREADING OF GEODESICS
2. Geodesics and Sectional Curvature
Fix v, w ∈ Tp M with 'v' = 'w' = 1 and let v(s) be a curve on Tp M such that
v(0) = v and v̇(0) = w.
• The rays t &→ tv(s) that start at 0 ∈ Tp M deviate from t &→ tv with velocity
' '
'
'∂ '
'
' ' tv(s)' = |tv| = t.
' ∂s s=0
'
• The rays t &→ expp (tv(s)) that start at p ∈ M deviate from t &→ γ(t) =
expp (tv) with velocity
' '
'
'∂ '
'
' ' expp (tv(s))' = |tv| = t.
'
' ∂s s=0
Define the parametrized surface f (s, t) := expp (tv(s)) and set
∂ ''
J(t) :=
' f (s, t).
∂s s=0
With this setting we are interested in finding |J(t)|.
Since for s fixed the curve t &→ f (s, t) is a geodesic, we have that
Therefore,
$
%
D D ∂f
D D ∂f
∂f ∂f ∂f
−
=R
,
∂t ∂s ∂t
∂s ∂t ∂t
∂s ∂t ∂t
%
$
D D ∂f
∂f ∂f ∂f
=R
,
∂t ∂t ∂s
∂s ∂t ∂t
$
%
D D ∂f
∂f
∂f
∂f
(0, t) + R
(0, t),
(0, t)
(0, t) = 0
∂t ∂t ∂s
∂t
∂s
∂t
D2
J(t) + R (γ̇(t), J(t)) γ̇(t) = 0
∂t2
D
dt
(
∂f
∂t
)
= 0.
Definition 2.1 (Jacobi Field). Let γ : [0, 1] → M be a geodesic in M . A vector
field J along γ is said to be a Jacobi field if it satisfies the equation
D2 J
(t) + R(γ̇(t), J(t))γ̇(t) = 0,
dt2
for all t ∈ [0, 1].
From now on let us write J ! for ' DJ
dt .
∂ '
We have shown that J(t) := ∂s ' f (s, t) is a Jacobi field along γ. This is not
s=0
a coincidence, in fact, the following result holds
Proposition 2.2. Let γ : [0, 1] → M be a curve in M such that γ(0) = p and
γ̇(0) = v. If J is a Jacobi field along γ with J(0) = 0 and w := J ! (0), then
∂ ''
J(t) =
' f (s, t)
∂s s=0
where f (s, t) = expp (tv(s)) for v(s) curve in Tp M satisfying v(0) = v and v ! (0) = w
Going back to studying the spreading of the geodesics, we have the following
theorem:
SPREADING OF GEODESICS
3
Theorem 2.3. If v, w ∈ Tp M form an orthonormal basis of the plane σ and J is
the Jacobi field satisfying J(0) = 0 and J ! (0) = w, then
|J(t)| = t −
Kp (σ) 3
t + r(t)
6
with r(t)/t3 → 0 as t → 0.
Proof. Let us compute the first five terms in the Taylor expansion of |J(t)|2 =
#J, J$ (t) at t = 0.
(1) Since J(0) = 0, #J, J$ (0) = 0.
!
!
(2) #J, J$ = 2 #J, J ! $. Then, #J, J$ (0) = 0.
!!
(3) #J, J$ = 2(#J ! , J ! $ + #J, J !! $). Thus, since 'J ! (0)' = 'w' = 1, we have
!!
#J, J$ (0) = 2.
!!!
(4) #J, J$ = 6 #J ! , J !! $ + 2 #J, J !! $. Therefore, since J(0) = 0 and J !! (0) =
!!!
−R(γ̇(0), J(0))γ̇(0), we get #J, J$ (0) = 0.
!!!!
(5) Finally, #J, J$ = 6 #J !! , J !! $ + 8 #J ! , J !!! $ + 2 #J, J !!! $. Using that J(0) = 0
!!!!
!!
and J (0) = 0, we get #J, J$ (0) = 8 #J ! , J !! $ (0).
D
!!
We know that J + R(γ̇, J)γ̇=0. Then J !!! + dt
R(γ̇, J)γ̇=0, or what is the
!!!
!
same, J + R(γ̇, J )γ̇ = 0. Therefore, since J !!! (0) = −R(v, w)v, and v, w
!!!!
form an orthonormal basis, we conclude that #J, J$ (0) = −8Kp (σ).
We have shown that
Kp (σ) 4
|J(t)|2 = t2 −
t + r(t)
3
where r(t)/t4 → 0 as t → 0. The result follows.
!
Since the rays in Tp M spread with velocity t and geodesics in M spread with
K (σ)
velocity t − p6 t3 we have shown that
• If Kp (σ) > 0, the geodesics t &→ expp (t(v(s))) spread slower than the rays
t &→ tv.
• If Kp (σ) < 0, the geodesics t &→ expp (t(v(s))) spread faster than the rays
t &→ tv.