Parallel Transport
Outline
1. Vector Fields Along Curves
Let S be a surface, and let γ : [a, b] → S be a curve on S. A vector field along γ is a
function that assigns to each point of γ a tangent vector to the surface at that point. That is,
a vector field along γ is a smooth function v : [a, b] → R3 with the property that v(t) ∈ Tγ(t) S
for all t ∈ [a, b].
For example, the unit tangent vector t to a curve is a vector field along the curve, as is
the tangent normal vector g. In general, any vector field along a curve can be written as
v = v1 t + v2 g
where v1 (t) and v2 (t) are real-valued functions.
2. Parallel Fields
Let γ be a curve on a surface S, and let v(t) be a vector field along γ. We say that v is
parallel along γ if the derivative v̇(t) of v(t) is normal to the surface at each point. For
example:
• If γ is a curve on a plane, then v(t) is parallel if and only if it is constant.
• If γ is a geodesic, then the unit tangent vector field t is parallel along γ, as is the tangent
normal vector field g. Indeed, a vector field v is parallel along a geodesic if and only if
v = At + B g
for some constants A and B. Such a vector field has constant length, and makes a
constant angle with the geodesic curve.
In general, the vectors of a parallel vector field must have constant length, but they need not
make a constant angle with a curve γ.
Given a curve γ on a surface and a starting vector v0 , there exists a unique parallel vector
field v(t) along γ with the given starting vector. This is called the parallel transport of v0
along γ.
3. Geodesic Curvature
Let γ be a unit speed curve, let v(t) be a nonzero parallel vector field along γ, and let ϕ(t)
be the counterclockwise angle from the tangent vector t to v(t). That is, suppose that
v(t) = (v cos ϕ)t + (v sin ϕ)g.
Then the geodesic curvature of γ is the negative of the derivative of ϕ:
κg = −ϕ̇
See Proposition 13.6.1 on pg. 362 in the book for a simple proof of this statement.
The minus sign comes from the fact that we are measuring the angle from the tangent
vector t to the vector v(t). Since the vectors v(t) are “parallel” and it’s actually t that’s
turning, it might make more sense to measure the angle from v(t) to t, in which case the
minus sign would disappear.
4. Holonomy
Now let S be a surface, and let γ : [0, L] → S be a closed curve on S. If v(t) is a parallel
vector field along γ, then the initial vector v(0) and the final vector v(L) may be different. In
this case, the holonomy around γ, denoted hγ , is defined to be the counterclockwise angle
from v(0) to v(L).
Let ϕ(t) denote the counterclockwise angle from t to v(t). Then the holonomy around γ
is given by the formula
Z L
Z
hγ = ϕ(L) − ϕ(0) =
ϕ̇ dt = − κg ds
(mod 2π).
γ
0
Since hγ is only defined modulo 2π, we can write this as
Z
hγ = 2π − κg ds
(mod 2π).
γ
Conceptually, hγ is the total amount that the vector v(t) turns as it goes around the curve γ.
5. Holonomy and the Gauss Map
Let S be a surface, and let G : S → S 2 be the Gauss map on S.
1. If γ : [a, b] → S is a closed curve on S, then its image G ◦ γ is a closed curve on the
sphere.
2. If v(t) is a vector field along γ, then v(t) is also a vector field along G ◦ γ.
3. If v(t) is a parallel along γ, then v(t) is also parallel along G ◦ γ.
4. The holonomy hγ of γ is the same as the holonomy hG◦γ of G ◦ γ.
For any simple closed curve on the sphere, the holonomy is the same as the area inside the
curve. Combining this with the last observation above, we obtain the Gauss-Bonnet Theorem
for simple closed curves:
Theorem. Let γ be any simple closed curve on a surface S, and suppose that γ bounds a
simply-connected region. Then the total Gaussian curvature over this region is equal to
the holonomy around γ:
ZZ
K dA = hγ
(mod 2π).
int(γ)
As written, this theorem is modulo 2π, but it is possible to state an exact version. Specifically,
ZZ
Z
K dA = 2π − κg ds
int(γ)
where κg is the geodesic curvature of γ.
γ