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Let gamma be a straight line in a surface M. How can we prove that gamma is a geodesic?

ALl I note is that a geodesic on a surface M is a unit speed curve on M with geodesic curvature = 0 everywhere.

Update: To making it not look like the question is a tautology, check out this:

http://www.physicsforums.com/showthread.php?t=407105

I'm trying to fill in the gaps and understand the argument for proving this theorem. Thanks

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..what do you mean by 'straight line'? Is $M$ embedded in $\mathbb R^n$? –  Berci Oct 8 '12 at 0:43
    
straight line = zero curvature –  mary Oct 8 '12 at 1:17
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but that's the definition of geodesic, too, no? –  Berci Oct 8 '12 at 1:22
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@mary: no. Tangent lines don't lie in $M$, they lie in some tangent space. I don't see any way of answering this question that isn't tautologous. The correct definition of "straight line" is geodesic. –  Qiaochu Yuan Oct 8 '12 at 2:27
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@Berci: Yes and no. The definition of a geodesic in $\mathbb{R}^3$ is that $\kappa = 0$ (what the OP refers to as "zero curvature"). The definition of a geodesic in $M$ is $\kappa_g = 0$ ("zero geodesic curvature"). –  Jesse Madnick Oct 10 '12 at 7:00
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up vote 3 down vote accepted
+50

Use the formula $$\kappa^2 = \kappa_g^2 + \kappa_n^2.$$

Here, $\kappa_g$ is the geodesic curvature, $\kappa_n$ is the normal curvature, and $\kappa$ is (unfortunately) just called the "curvature" (it is the $\kappa$ that appears in the Frenet-Serret Formulas).

A straight line has $\kappa = 0$, so....

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so then if kappa = 0 and kappag = 0, then the result is that kappan = 0 so everything is zero. But how can you incorporate this argument in an iff statement –  mary Oct 11 '12 at 4:02
    
My point is that if $\kappa = 0$, then $\kappa_g = 0$. This says exactly that straight lines are geodesics. –  Jesse Madnick Oct 11 '12 at 4:34
    
The converse is false: geodesics in $M$ are not necessarily straight lines. –  Jesse Madnick Oct 11 '12 at 4:34
    
how do we know that kappag = 0? –  mary Oct 11 '12 at 7:57
    
Why is it if kappa=0 then Kappasubg =0? –  mary Oct 11 '12 at 18:07
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