Page 397 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes In other words, the principal normal n must be parallel to the normal N to the surface along the

curve segment from p to q.
If C is parameterized by arc length, this means that the acceleration must be normal to the
surface.
It is then natural to define geodesics as those curves such that k = 0 everywhere on their domain
g
of definition.
Actually, there is another way of defining geodesics in terms of vector fields and covariant
derivatives, but for simplicity, we stick to the definition in terms of the geodesic curvature.

Definition 1. Given a surface X :   E , a geodesic line, or geodesic, is a regular curve
3
C : I  E on X, such that k (t) = 0 for all t  I.
3
g


Notes By regular curve, we mean that C(t)  0 for all t  I, i.e., C is really a curve, and
not a single point.
Physically, a particle constrained to stay on the surface and not acted on by any force, once set in
motion with some non-null initial velocity (tangent to the surface), will follow a geodesic
(assuming no friction).

Since k = 0 if the principal normal n to C at t is parallel to the normal N to the surface at X(u(t),
g

v(t)), and since the principal normal n is a linear combination of the tangent vector C(t) and

the acceleration vector C(t), the normal N to the surface at t belongs to the osculating plane.

Since the tangential part of the curvature at a point is given by

   

' 
' 
k g g n    u   1 ij u u X    u   2 ij u u X ,
"
'
'
2 
1 
"
j
u

i
j
i
v

 i 1,2   i 1,2 


 j 1,2   j 1,2 


the differential equations for geodesics are
1 
"
'
'
u   1 ij u u  0,
i
j
i 1,2

j 1,2

'
2 
"
'
u   2 ij u u  0,
j
i
i 1,2

j 1,2

or more explicitly (letting u = u and v = u ),
2
1
2
2
u"   1 11 (u')  2 1 12 u'v'  1 22 (v')  0,
2
2
v"   2 11 (u')  2 2 12 u'v'  2 22 (v')  0.
In general, it is impossible to find closed-form solutions for these equations.
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