Antipodal Points and Diameter of a Sphere

    Received 04 November 2018

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

    Author(s): Podobryaev A. V.

    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. V., Antipodal Points and Diameter of a Sphere, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  579-581
    DOI:10.20537/nd180410


    Download File
    PDF, 203.95 Kb

    References
    [1] Agrachev, A. A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., 87, Springer, Berlin, 2004  crossref  mathscinet  zmath
    [2] Berger, M., “Les variétés riemanniannes homogènes normales simplement connexes à courbure strictement positive”, Ann. Scoula Norm. Sup. Pisa, 15:3 (1961), 179–246  mathscinet  zmath
    [3] Nikonorov, Yu. G., “For a Geodesic Diameter of Surfaces with Isometric Involution”, Trudy Rubtzovsk. Industrialn. Inst., 9 (2001), 62–65 (Russian)  zmath
    [4] Nikonorov, Y. G. and Nikonorova, Y. V., “The Intrinsic Diameter of the Surface of a Parallelepiped”, Discrete Comput. Geom., 40:4 (2008), 504–527  crossref  mathscinet  zmath
    [5] Novikov, S. P. and Taimanov, I. A., Modern Geometric Structures and Fields, Grad. Stud. Math., 71, AMS, Providence, R.I., 2006, xx+633 pp.  crossref  mathscinet  zmath
    [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



    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License