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Math Help - Differential Geometry: Straight lines as shortest connections

  1. #1
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    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?
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  2. #2
    Super Member Rebesques's Avatar
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    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.
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  3. #3
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    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)
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  4. #4
    Super Member Rebesques's Avatar
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    The Cauchy-Schwartz inequality is pretty basic.
    I don't think you need to prove it in every new course.
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