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Unit 31: Joachimsthal's Notations
Nevertheless, from the theory of ordinary differential equations, the following lemma showing Notes
the local existence of geodesics can be shown :
Lemma 1: Given a surface X, for every point p = X(u, v) on X, for every non-null tangent vector
v T (u,v) (X) at p, there is some > 0 and a unique curve : ] , [ E on the surface X, such
3
that is a geodesic, (0) = p, and (0) = v.
To emphasize that the geodesic depends on the initial direction v, we often write (t, v )
instead of (t).
The geodesics on a sphere are the great circles (the plane sections by planes containing the center
of the sphere).
More generally, in the case of a surface of revolution (a surface generated by a plane curve
rotating around an axis in the plane containing the curve and not meeting the curve), the
differential equations for geodesics can be used to study the geodesics.
Example: The meridians are geodesics (meridians are the plane sections by planes through
the axis of rotation: they are obtained by rotating the original curve generating the surface).
Also, the parallel circles such that at every point p, the tangent to the meridian through p is
parallel to the axis of rotation, is a geodesic.
It should be noted that geodesics can be self-intersecting or closed. A deeper study of geodesics
requires a study of vector fields on surfaces and would lead us too far.
Technically, what is needed is the exponential map, which we now discuss briefly.
The idea behind the exponential map is to parameterize locally the surface X in terms of a map
from the tangent space to the surface, this map being defined in terms of short geodesics.
More precisely, for every point p = X(u, v) on the surface, there is some open disk B of center
(0, 0) in (recall that the tangent plane T (X) at p is isomorphic to ), and an injective map
2
2
p
exp : B X(),
p
such that for every v B with v 0,
exp (v) (1,v),
p
where (t,v) is the unique geodesic segment such that (0,v) p and (0,v) ' v. Furthermore,
for B small enough, exp is a diffeomorphism. It turns out that exp (v) is the point q obtained
p p
by laying off a length equal to v along the unique geodesic that passes through p in the
direction v.
3
2
Lemma 2: Given a surface X : E , for every v 0 in , if
( ,v) :] , [ E 3
is a geodesic on the surface X, then for every > 0, the curve
( , v) :] / , / [ E 3
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