Lecture Notes for 120 - UCLA Department of Mathematics

7.2. UNPARAMETRIZED GEODESICS 159
(4) Let q (t) be a unit speed geodesic on a surface in space. Show that
0 = apple g
apple = apple n
⌧ = ⌧ g
where apple and ⌧ are the curvature and torsion of q (t) as a space curve.
7.2. Unparametrized Geodesics
It is often simpler to find the unparametrized form of the geodesics, i.e., in
agivenparametrizationtheyareeasiertofindasfunctionsu (v) or v (u) . We
start with a tricky characterization showing that one can characterize geodesics
without referring to the arclength parameter. The idea is that a regular curve can
be reparametrized to be a geodesic if and only if its tangential acceleration ¨q I is
tangent to the curve.
Lemma 7.2.1. A regular curve q (t) =q (u (t) ,v(t)) can be reparametrized as
a geodesic if and only if
✓
dv d 2 ◆
u
dt dt 2 + u ( ˙q, ˙q) = du ✓ d 2 ◆
v
dt dt 2 + v ( ˙q, ˙q) .
Proof. Let s correspond to a reparametrization of the curve. When switching
from t to s we note that the left hand side becomes
✓
dv d 2 ◆
u
dt dt 2 + u ( ˙q, ˙q) = dv ✓ d 2 ✓
u dq
dt dt 2 + u dt , dq ◆◆
dt
= ds dv d 2 ✓ ◆ 2
s du ds
dt ds dt 2 ds + d 2 ✓
u ds
dt ds 2 + u dq
dt ds , ds ◆ !
dq
dt ds
= ds dv d 2 ✓ ◆ 2
s du ds
dt ds dt 2 ds + d 2 ✓ ◆ 2 ✓
u ds
dt ds 2 + u dq
dt ds , dq ◆ !
ds
= ds d 2 ✓
s dv du ds
dt dt 2 ds ds + dt
◆ 3
dv
ds
✓ d 2 u
ds 2 +
with a similar formula for the right hand side. Here the first term
ds d 2 s dv du
dt dt 2 ds ds
✓ dq u
ds , dq
ds
is the same on both sides, so we have shown that the equation is actually independent
of parametrizations. In other words if it holds for one parametrization it holds
for all reparametrizations.
If q is a geodesic then the formula clearly holds for the arclength parameter.
Conversely if the equation holds for some parameter then it also holds for the
arclength parameter. Being parametrized by arclength gives us the equation
I ˙q, ¨q I = ⇥ du
dt
dv
dt
⇤ apple g uu
"
g uv
g vu g vv
d 2 u
dt
+ 2 u ( ˙q, ˙q)
d 2 v
dt
+ 2 v ( ˙q, ˙q)
#
=0
◆◆