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

180 Chapter 4. General theory of surfaces
4.19.3 Geodesic lines
Definition 4.19.1. Let S be a surface. A parameterized curve ρ : I → S is called a
geodesic line, or simply, a geodesic if, at each point, its geodesic curvature vanishes.
Remark. According to the formula (4.19.12), the geodesic curvature of a curve can be
written as
kg = (τ, τ ′ , n).
On the other hand, from the first formula of Frenet, τ × τ ′ = kβ, therefore, kg vanishes at
a point of a curve on the surface if and only if the binormal of the curve is perpendicular
on the normal of the surface at that point or, in other words, the geodesic curvature
vanishes if and only if the normal of the surface is contained in the osculating plane.
Thus, the geodesic lines are exactly those lines on the surface for which the osculating
plane at each of their points contain the normal of the surface at that point. In fact, in
many books, this property is taken as the definition of the geodesics.
Another remark that should be made is that since, as we noticed earlier, the geodesic
curvature is the signed curvature of the projection of the curve on the tangent plane,
we can say that the geodesics are those curves which project on each tangent plane of
the surface after a straight line. In this sense, we might say that the geodesics are the
straightest lines on the surfaces.
The formula 4.19.9 leads to
Theorem 4.5. The differential equation of the geodesic lines is
Γ 2 11u′3 + � 2Γ 2 12 − Γ1 � �
′2 ′ 2
11 u v + Γ22 − 2Γ 1 �
′ ′2 1
12 u v − Γ22v ′3 + u ′ v ′′ − u ′′ v ′ = 0. (4.19.11)
Also, from (4.19.10) we can deduce that
Theorem 4.6. The geodesic lines verify the system of differential equations
⎧
⎪⎨ u
⎪⎩
′′ + u ′2Γ1 11 + 2u′ v ′ Γ1 12 + v′2Γ1 22 = 0
. (4.19.12)
= 0
v ′′ + u ′2 Γ 2 11 + 2u′ v ′ Γ 2 12 + v′2 Γ 2 22
There is, apparently, a contradiction between the previous two theorems, as the first
claim that the geodesics are solutions of a single equations, while the second – that they
are a solution of a system of different equation. In fact, however, the two equations of the
system (4.19.12) are not independent, because we impose the geodesics to be naturally
parameterized, therefore, between the functions u and v there is an extra relation.
The existence (locally, at least) of geodesics passing through a point of a surface and
having at that point a given tangent vector is a consequence of standard results in the
theory of ordinary differential equations. Finding geodesics, is, usually, a very delicate
business and can be done explicitly only in special situations.