Blaga P. Lectures on the differential geometry of - tiera.ru

Examples of geodesics
4.19. Geodesics 181
The geodesics of the plane. By the geometrical interpretation of the geodesics, as
the curves that project on the tangent plane on straight lines, we deduce immediately
that the geodesics of the plane are the straight lines and only them. On the other hand,
by using cartesian coordinates, we fined immediately that the Christoffel’s coefficients
vanish identically, therefore the equations of geodesics become
⎧
⎪⎨ u
⎪⎩
′′ = 0
v ′′ ,
= 0
which lead to u(s) = a1s + b1, v(s) = a2s + b2, i.e., again, the geodesics are straight lines.
The geodesics of the sphere. The geodesics of the sphere can be found very easily
from their interpretation as being those curves for which the osculating plane contains
the normal to the surface. As we know, in the case of the sphere all the normals are
passing through the center of the sphere, therefore the osculating planes should pass, all
of them through the center, which leads immediately to the idea that the geodesics are
arcs of great circles of the sphere.
Also, using a standard parameterization of the sphere (with spherical coordinates),
we can find (do that!) the equations of the geodesics as:
⎧
⎪⎨ ¨θ − sin θ cos θ ˙ϕ
⎪⎩
2 = 0
.
¨ϕ + 2 cot θ˙θ ˙ϕ = 0
We assume that the explicit equation of the geodesic is θ = θ(ϕ). Then
˙θ = dθ
ds
¨θ = d
ds
dθ
= · ˙ϕ,
dϕ
� �
dθ
· ˙ϕ +
dϕ
dθ dθ
¨ϕ =
dϕ dϕ · ¨ϕ + d2θ dϕ2 · ˙ϕ2 .
Therefore, the first equation of the system becomes:
On the other hand, from the first equation,
¨ϕ · dθ
dϕ + d2 θ
dϕ 2 · ˙ϕ2 − sin θ cos θ · ˙ϕ 2 = 0.
¨ϕ = −2 cot θ˙θ · ˙ϕ = −2 cot θ dθ
dϕ · ˙ϕ2 ,