Geodesic
Antipodal Points and Diameter of a Sphere
Russian journal of nonlinear dynamics, Tome 14 (2018) no. 4, pp. 579-581 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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, Berger sphere, cut locus, geometric control theory.
Mots-clés : $SU_2$, antipodal points 
@article{ND_2018_14_4_a9,
     author = {A. V. Podobryaev},
     title = {Antipodal {Points} and {Diameter} of a {Sphere}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {579--581},
     year = {2018},
     volume = {14},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ND_2018_14_4_a9/}
}
TY  - JOUR
AU  - A. V. Podobryaev
TI  - Antipodal Points and Diameter of a Sphere
JO  - Russian journal of nonlinear dynamics
PY  - 2018
SP  - 579
EP  - 581
VL  - 14
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ND_2018_14_4_a9/
LA  - en
ID  - ND_2018_14_4_a9
ER  - 
%0 Journal Article
%A A. V. Podobryaev
%T Antipodal Points and Diameter of a Sphere
%J Russian journal of nonlinear dynamics
%D 2018
%P 579-581
%V 14
%N 4
%U http://geodesic.mathdoc.fr/item/ND_2018_14_4_a9/
%G en
%F ND_2018_14_4_a9
A. V. Podobryaev. Antipodal Points and Diameter of a Sphere. Russian journal of nonlinear dynamics, Tome 14 (2018) no. 4, pp. 579-581. http://geodesic.mathdoc.fr/item/ND_2018_14_4_a9/

[1] Agrachev, A. A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., 87, Springer, Berlin, 2004 | DOI | MR | Zbl

[2] Berger, M., “Les variétés riemanniannes homogènes normales simplement connexes à courbure strictement positive”, Ann. Scoula Norm. Sup. Pisa, 15:3 (1961), 179–246 | MR | Zbl

[3] Nikonorov, Yu. G., “For a Geodesic Diameter of Surfaces with Isometric Involution”, Trudy Rubtzovsk. Industrialn. Inst., 9 (2001), 62–65 (Russian) | Zbl

[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 | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[6] Mat. Zametki, 103:5 (2018), 779–784 (Russian) | DOI | DOI | MR | Zbl

[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 | DOI | MR | Zbl