Geodesics and straight lines on a surface

  • #1

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.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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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.
 

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