An \(n\)-dimensional flat torus in \(S^{2n-1}\) whose extrinsic diameter is equal to \(\pi\) (Q1373457)

The author investigates isometries immersions of the flat \(n\)-torus \(T^n\) into \(S^{2n-1}\). It is shown that if the diameter of the immersion \(f\) equals \(\pi\), then the immersion \(f\) is congruent to the standard immersion of \(T^n\) into \(S^{2n-1}\). For \(n=2\), this result was already obtained by \textit{K. Enomoto}, \textit{Y. Kitagawa} and \textit{J. L. Weiner} [Proc. Am. Math. Soc. 124, 265-268 (1996; Zbl 0856.53043)]. The author also proposes to investigate the following open problem: does there exist an isometric immersion \(f\) of \(T^n\) into \(S^{2n-1}\) such that the extrinsic diameter is less then \(\pi\)?