Geodesics and straight lines on a surface

This theorem does not apply to straight lines on surfaces.In summary, the shortest distance between two points on a surface M is a geodesic if the curve is unit speed. However, it is not necessary for an arbitrary straight line to be unit speed and this theorem does not apply to straight lines on surfaces.
  • #1
SNOOTCHIEBOOCHEE
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Homework Statement



Let [tex]\gamma[/tex] be a stright line in a surface M. Prove [tex]\gamma[/tex] is a geodeisc



The Attempt at a Solution



In a plane we know a straight line is the shortest distance between two point. I am not sure if this applies to straight lines on a surface.

Further more, there is a theorem that says that if [tex]\gamma[/tex] is a unit speed curve and the shortest distance between two points P= [tex]\gamma (a)[/tex] and [tex]Q=\gamma (b)[/tex]then it is a geodesic.

But i do not know how to show some arbitrary straight line is unit speed or if this approach is even valid.

Any help appreciated.
 
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  • #2
You know that a straight line is the shortest distance between two points in the Euclidean space that contains the surface. An arbitrary straight line doesn't have to be unit parameterized, but you can certainly parameterize it to be unit length.
 

1. What is a geodesic?

A geodesic is the shortest path between two points on a curved surface, such as a sphere or a curved plane. It is the equivalent of a straight line on a flat surface.

2. How is a geodesic different from a straight line on a surface?

While a straight line on a flat surface is the shortest distance between two points, a geodesic on a curved surface is the shortest distance between those same points, but taking into account the curvature of the surface.

3. Can a geodesic ever be a curve?

Yes, a geodesic can be a curve on a curved surface. This is because the shortest path between two points on a curved surface may not necessarily be a straight line, but rather a curve that follows the curvature of the surface.

4. How are geodesics and straight lines related to each other?

Geodesics are the equivalent of straight lines on a curved surface. They follow the same principles of being the shortest path between two points, but take into account the curvature of the surface.

5. What is the importance of geodesics in mathematics and science?

Geodesics play a crucial role in understanding and describing the geometry of curved surfaces, such as the Earth's surface. They are also important in fields such as physics, where they are used to calculate the shortest path of light rays in curved space-time and in engineering, where they are used in designing curved structures and surfaces.

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