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Thread: Distance regarding 5-Sphere

  1. #1
    Junior Member
    Oct 2012

    Distance regarding 5-Sphere


    I've got to answer two questions:

    a) Let $S_1 , S_2 \in \mathbb{S}^5$ be two three dimensional compact totally geodesic submanifolds. Can $d(S_1,S_2)\geq \frac{\pi}{2}$ hold?

    b) Is each minimising curve a geodesic?

    Note: By $d(.,.)$ the distance is meant.

    My thoughts:
    a) curvature of the sphere is 1. The 1-sphere has a circumference of $\pi$. How to calculate the circumference of $\mathbb{S}^5$ Totally geodesic means that the second fundamental form disappears. We have that each geodesic in $S_1$ or $S_2$ is a geodesic in $\mathbb{S}^5$. How do I continue here in order to get a contradiction or is the claim right?

    b) I've seen the expression "minimising geodesic" on wikipedia. Is there an example of a minimising curve that is NOT a geodesic? Do you know one? How to find one? Or is the claim right?
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  2. #2
    Super Member Rebesques's Avatar
    Jul 2005
    My house.

    Re: Distance regarding 5-Sphere

    What tools are we at liberty to use?
    Has the instructor spoken to you, say, about the Gauss-Bonnet theorem?
    Last edited by Rebesques; Jul 8th 2014 at 05:15 PM.
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