Mathematical problems of nonlinearity
Antipodal Points and Diameter of a Sphere
A. V. Podobryaev
A. K. Ailamazyan Program Systems Institute of RAS, ul. Petra-I 4a, Veskovo, Pereslavl district, Yaroslavl region, 152021 Russia
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.
diameter, $SU_2$, Berger sphere, antipodal points, cut locus, geometric control theory
|Russian Science Foundation
|This work was supported by the Russian Science Foundation under grant 17-11-01387 and was carried out at Ailamazyan Program Systems Institute of the Russian Academy of Sciences.
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MSC: 53C20, 53C22, 53C30, 49J15
A. V. Podobryaev, “Antipodal Points and Diameter of a Sphere”, Nelin. Dinam., 14:4 (2018), 579–581
Citation in format AMSBIB
\by A. V. Podobryaev
\paper Antipodal Points and Diameter of a Sphere
\jour Nelin. Dinam.
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