DIFFERENTIAL GEOMETRY COURSE

21. *GEODESICS*

*Explanation 182 :* Suppose a car is riding over a surface, and gravitation is pulling it in the direction of -* N*. If the driver doesn't need to
turn the wheel, the car is riding over a geodesic.

This is the case if and only if its traject curve

Another description is the following.

Given a curve on a surface. The tangent planes in the points of the curve are enveloping a developable surface, whereon this curve is lying. When we develop this last surface, the image of the curve on the
surface is a planar curve, the socalled trace curve. In other words: draw the curve with wet ink on the surface and roll the developable surface that the tangent planes are enveloping over the floor in such
a way that the curve leaves a print on the floor. The print is the trace curve.

If the trace curve is a straight line, we call the original curve a geodesic of the surface. The shortest traject over the surface between two points on the surface is always a geodesic.
We call the curvature of the trace curve geodetic curvature.

*Example 183:* Draw the trace curve of a great circle on the sphere (the corresponding developable surface that the tangent planes are enveloping is here a right circular cylinder),
and draw also the trace curve of a little circle (the corresponding developable surface that the tangent planes are enveloping is then a right circular cone).

So which curves are the geodesics of the sphere?

The next thing we have to do is deducing the equations we need to calculate the geodesics of a surface.

*Definition 184:* The geodesic curvature of a curve * x*(s) on a surface is the length of the tangential component of the curvature vector

κ_{geod} = ||__x__^{..} - (__x__^{..}).* N*|| = |κ| ||

where κ is the curvature of * x*(s),

We also have κ

*Definition 185:* We call a curve on the surface a geodesic of this surface if the geodesic curvature is 0 in all points of the curve.

*Proposition 186:* Define the Christoffel symbols Γ_{ij}^{k} and γ_{ij}^{k} by
__x___{uiuj} = Γ_{ij}^{1} __x___{u1} + Γ_{ij}^{2} __x___{u2} +
h_{ij} * N*,

γ

Then we find the geodesics

Γ_{11}^{k} (u_{1}^{.})^{2} + 2 Γ_{12}^{k} (u_{1}^{.} u_{2}^{.}) +
Γ_{22}^{k} (u_{2}^{.})^{2} + u_{k}^{..} = 0 (k=1,2).

*Proof:* Check by calculation that
__x__^{..} = (__x___{u1u1} u_{1}^{.} +
__x___{u1 u2} u_{2}^{.}) u_{1}^{.} + __x___{u1} u_{1}^{..} +
(__x___{u1u2} u_{1}^{.} +
__x___{u2 u2} u_{2}^{.}) u_{2}^{.} + __x___{u2} u_{2}^{..}.

The tangential component of __x__^{..} must be 0. This gives the equations.

*Proposition 187:* Christoffel symbols are intrinsic variables and the concept geodesic is invariant under bending, because

2 γ_{ij}^{k} = (δ/δu_{i}) a_{j k} + (δ/δu_{j}) a_{i k} - (δ/δu_{k}) a_{i j}.

*Proof:* From a_{i k} = __x___{ui} **.** __x___{uk} we get (δ/δu_{j}) a_{i k} =
γ_{ij}^{k} + γ_{kj}^{i}.

*Remark 188:* The theory of differential equations learns that through any given point on a surface there goes exactly one geodesic in every direction.
And through any two points on the surface there goes a geodesic that is the shortest traject over the surface between these two points.

*Problem 189:* Prove that the geodesics of the cylinder (a cos(v), a sin(v), u) are: the right circular helices, the circles and the rules.

*Problem 190:* Determine the geodesics on the paraboloid surface of revolution (u cos(v), u sin(v), (1/2)u^{2}).

*Problem 191:* Determine the geodesics on the right circular cone.

*Problem 192:* The principal normal of a geodesic is also normal on the surface, so the osculating plane contains the normal on the surface.

Hence prove that an asymptotic line that is also a geodesic must be a straight line (see 142).

Are the circles of latitude of an arbitrary surface of revolution geodesics? And the meridian curves?