Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics]
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Nelin. Dinam., 2018, Volume 14, Number 4, Pages 579–581 (Mi nd632)  

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

Abstract: 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

Funding Agency Grant Number
Russian Science Foundation 17-11-01387
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.


DOI: https://doi.org/10.20537/nd180410

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MSC: 53C20, 53C22, 53C30, 49J15
Received: 04.11.2018
Revised: 01.12.2018
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Citation: A. V. Podobryaev, “Antipodal Points and Diameter of a Sphere”, Nelin. Dinam., 14:4 (2018), 579–581

Citation in format AMSBIB
\Bibitem{Pod18}
\by A. V. Podobryaev
\paper Antipodal Points and Diameter of a Sphere
\jour Nelin. Dinam.
\yr 2018
\vol 14
\issue 4
\pages 579--581
\mathnet{http://mi.mathnet.ru/nd632}
\crossref{https://doi.org/10.20537/nd180410}
\elib{https://elibrary.ru/item.asp?id=36686075}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85061714180}


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