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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
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References:

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