# Differential geoemtry tangent lines parallel proof

• hlin818

## Homework Statement

Prove that a(s) is a straight line if and only if its tangent lines are all parallel.

## Homework Equations

Frenet serret theorem

## The Attempt at a Solution

I'm confused on the direction "if the tangent lines are parallel then a(s) is a straight line".

Assume all the tangent lines of a(s) are parallel. So the tangent vector T is the same for all points xo on the curve a(s) and the values of T(s) of any two points on the curve are parallel. Thus T(s) is constant, and T'(s)=0 which implies that the curvature is zero, and thus a(s) must be a straight line.

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so you know the direction fo the tanget vector at all times, eg.

T(t) = a.g(t)

where a is a constant vector and g is a scalar function. Think about integrating this to get the original curve and/or the effect on other parameters, curvature etc.

Is the way I did it incorrect?

hlin818 said:
Is the way I did it incorrect?

It seems fine to me.

Thanks!