Diameter of the Berger sphere. (English. Russian original) Zbl 1402.58008

Math. Notes 103, No. 5, 846-851 (2018); translation from Mat. Zametki 103, No. 5, 779-784 (2018).
Let \(d(M)\) be the diameter of a Riemannian manifold \(\mathcal{M}=(M,g)\). Consider the special example in which \(M=\operatorname{SU}(2)=S^3\) is the sphere equipped with a left invariant metric with eigenvalues \((I_1,I_2,I_3)\). Assume \(I_1=I_2\). The author shows
1. \(d(I_1,I_2,I_3)=2\pi\sqrt{I_1}\) if \(I_1\leq I_3\).
2. \(2\pi\sqrt{I_3}\) if \(I_1\in (I_3,2I_3]\).
3. \(\frac{\pi I_1}{\sqrt{I_1-I_3}}\) if \(I_1>2I_3\).
Section 2 deals with the cut locus and the cut time and Section 3 gives the proof of the main theorem.


58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
Full Text: DOI arXiv


[1] O. Lablée, Spectral Theory in Riemannian Geometry (EMS, Zürich, 2015). · Zbl 1328.53001 · doi:10.4171/151
[2] Berger, M., No article title, Ann. Scuola Norm. Sup. Pisa (3), 15, 179, (1961)
[3] N. Eldredge, M. Gordina, and L. Saloff-Coste, Left-Invariant Geometries on SU(2) are Uniformly Doubling, https://arxiv.org/abs/1708.03021 (2017).
[4] Podobryaev, A. V.; Sachkov, Yu. L., No article title, J. Geom. Phys., 110, 436, (2016) · Zbl 1352.53044 · doi:10.1016/j.geomphys.2016.09.005
[5] A. A. Agrachev and Yu. L. Sachkov, Control Theory from Geometric Viewpoint (Fizmatlit, Moscow, 2004; Springer-Verlag, Berlin, 2004). · Zbl 1062.93001 · doi:10.1007/978-3-662-06404-7
[6] Bates, L.; Fassò, F., No article title, Int. Math. Forum, 2, 2109, (2007) · Zbl 1151.53348 · doi:10.12988/imf.2007.07190
[7] Sachkov, Yu. L., No article title, Mat. Sb., 197, 123, (2006) · doi:10.4213/sm1548
[8] S. P. Novikov and I. A. Taimanov, Modern Geometric Structures and Fields, in Graduate Studies in Mathematics (MTsNMO,Moscow, 2005; AMS, Providence, RI, 2006), Vol.71.
[9] S. G. Krantz and H. R. Parks, Implicit Function Theorem: History, Theory, and Applications (Birkhäuser, Basel, 2001). · Zbl 1012.58003
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