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    Antipodal Points and Diameter of a Sphere

    2018, Vol. 14, no. 4, pp.  579-581

    Author(s): Podobryaev A.

    We give an example of a Riemannian manifold homeomorphic to a sphere such that its diameter cannot be realized as a distance between antipodal points. We consider a Berger sphere, i.e., a three-dimensional sphere with Riemannian metric that is compressed along the fibers of the Hopf fibration. We give a condition for a Berger sphere to have the desired property. We use our previous results on a cut locus of Berger spheres obtained by the method from geometric control theory.
    Keywords: diameter, $SU_2$, Berger sphere, antipodal points, cut locus, geometric control theory
    Citation: Podobryaev A., Antipodal Points and Diameter of a Sphere, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  579-581
    DOI:10.20537/nd180410


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    References

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    [6] Podobryaev, A. V., “Diameter of the Berger Sphere”, Math. Notes, 103:5–6 (2018), 846–851  mathnet  crossref  mathscinet  zmath; Mat. Zametki, 103:5 (2018), 779–784 (Russian)  crossref  mathscinet  zmath
    [7] Podobryaev, A. V. and Sachkov, Yu. L., “Cut Locus of a Left Invariant Riemannian Metric on $SO(3)$ in the Axisymmetric Case”, J. Geom. Phys., 110 (2016), 436–453  crossref  mathscinet  zmath  adsnasa



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