## 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
Theorem.
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.

### MSC:

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

### References:

 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.