# Thread: Differential Geometry: Straight lines as shortest connections

1. ## Differential Geometry: Straight lines as shortest connections

I started this problem, but got stuck shortly after beginning
Basically, I started with the middle peice of math from the in/equation above.

Since v is a constant vector, I took it out the front of the integral and used the fundamental theorem of calculus such that the integral of the derivative collapses to γ(t) with the terminals b and a.

Subbing in the bounds, leads to γ(b) - γ(a), and given the equations in the equation, this simplies to v[Q-P], notice how this is different to the left hand side of the in/equality (PQ)v.

Is my method on the right track, or have I gone completely off the rails?

2. Use the Cauchy-Schwartz inequality:

$\frac{d\gamma}{dt}\cdot v\leq \left\vert\frac{d\gamma}{dt} \right\vert \vert v\vert =\left\vert\frac{d\gamma}{dt} \right\vert$

and integrate.

Ps. To show that straight lines are distance minimizers, choose $v=\frac{1}{|PQ|}PQ$ and conclude that the minimum of curve distances is attained if $\gamma$ is a straight line.

3. Hi, thanks for the reply.
Just curious as to know if there are any alternative methods, as I do not beleive Cauchy-Schwartz inequality has been covered in this unit (although in past years I have seen glimpses of it)

4. The Cauchy-Schwartz inequality is pretty basic.
I don't think you need to prove it in every new course.