Differential Geometry - Computer Graphics and Geometric Modeling - page 636

of the surface but seems to depend on the surface being imbedded in R 3 because we
used a normal vector to the surface. We shall see later in Section 9.16 that geodesics
really do not depend on any imbedding. In fact, they can be defined for any manifold,
not just a surface, and depend only on the metric coefficients. It is at that point of
abstraction that one sees why the preferred development of geodesics is via a global
notion of parallel vector fields and why the geodesic curvature is the secondary notion.
Of course, one will have to give a more intrinsic definition of parallel vector fields
since the definitions for the covariant derivative and parallel vector fields in this
section also depended on a normal vector.
As one final point about parallel vector fields, we address the question of their
existence and uniqueness.

9.10.18. Theorem. Let S be a surface in R 3 and let h : [a,b] Æ S be a curve.
Let v ΠT h(a) ( S ). Then there exists a unique parallel vector field X (t) along h with
X (a) = v .

Proof. The proof simply consists of analyzing the condition for a vector field to be
parallel. The condition is equivalent to finding a solution to first order differential
equations with initial condition v . See [Thor79].

An interesting consequence of the last theorem is

9.10.19. Corollary. Let S be a surface in R 3 and let g(s) be a geodesic in S . Then a
vector field X along g is parallel along g if and only if both the length of all the vectors
of X and the angle between them and g¢(s) is constant.

Proof. See [Thor79].

Furthermore, since parallel vector fields are completely defined by their initial
vector, one introduces the following useful terminology.

Definition. Let S be a surface in R 3 and let h : [a,b] Æ S be a curve. Let v Œ T h(a) ( S ).
Let X be the unique vector field along h that satisfies X (a) = v . If c Π[a,b], then X (c)
is called the parallel translate of v to h(c) along h.

We finish our introduction to geodesics on surfaces by defining one more impor-
tant map. It turns out that one can rephrase the result in Theorem 9.10.12 as follows:

9.10.20. Theorem. Let S be a surface in R 3 . If p ΠS , then there is an e p > 0 so that
for all v Œ T p ( S ) with | v | <e p there is a unique geodesic g :(-2,2) Æ S with g(0) = p
and g¢(0) = v .

Proof. Exercise 9.10.5.

Using the notation in Theorem 9.10.20 let

() =Π()

{ } Õ

() Up v T S v < e T S

p p p