Page 400 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 31: Joachimsthal's Notations

It can also be shown that the Gaussian curvature is expressed as follows: Notes

K = – 1  2 ( G) .
G  2
Polar coordinates can be used to prove the following lemma showing that geodesics locally
minimize arc length:
However, globally, geodesics generally do not minimize arc length.
For instance, on a sphere, given any two non-antipodal points p, q, since there is a unique great
circle passing through p and q, there are two geodesic arcs joining p and q, but only one of them
has minimal length.

Lemma 4: Given a surface X :   E , for every point p = X(u, v) on X, there is some  > 0 and
3
some open disk B of center (0, 0) such that for every q  exp (B ), for every geodesic  : ] – ,
p


[  E in exp (B ) such that (0) = p and (t ) = q, for every regular curve  : [0, t ]  E on X such
3
3
p
1
1

that (0) = p and (t ) = q, then
1
l (pq)  l (pq),
 
where l (pq) denotes the length of the curve segment  from p to q (and similarly for ).

Furthermore, l (pq) = l (pq) if the trace of  is equal to the trace of  between p and q.
 
As we already noted, lemma 4 is false globally, since a geodesic, if extended too much, may not
be the shortest path between two points (example of the sphere).
However, the following lemma shows that a shortest path must be a geodesic segment:
Lemma 5: Given a surface X :   E , let  : I  E be a regular curve on X parameterized by arc
3
3
length. For any two points p = (t ) and q = (t ) on , assume that the length l (pq) of the curve
0
1

segment from p to q is minimal among all regular curves on X passing through p and q. Then,
 is a geodesic.
At this point, in order to go further into the theory of surfaces, in particular closed surfaces, it is
necessary to introduce differentiable manifolds and more topological tools.
Nevertheless, we can’t resist to state one of the “gems” of the differential geometry of surfaces,
the local Gauss-Bonnet theorem.
The local Gauss-Bonnet theorem deals with regions on a surface homeomorphic to a closed disk,
whose boundary is a closed piecewise regular curve  without self-intersection.
Such a curve has a finite number of points where the tangent has a discontinuity.
If there are n such discontinuities p , . . . , p , let  be the exterior angle between the two tangents
1
i
n
at p . i
More precisely, if (t ) = p , and the two tangents at p are defined by the vectors
i
i
i

lim  '(t)   '_(t ) 0,

i
t i t , t t  i
and

lim  '(t)   ' (t )  0,
t i t , t t  i  i

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