On antipodes on a manifold endowed with a generic Riemannian metric.
The paper is related to some conjectures about antipodes formulated by \textit{H. Steinhaus} [Unsolved problems in geometry, New York: Springer-Verlag (1991; Zbl 0748.52001], for example: Is the sphere the only surface on which each point admits a single antipode such that the antipodal function is an involution? This question was solved by \textit{C. Vîlcu} [Geom. Dedicata 79, No. 3, 267--275 (2000; Zbl 0971.52003)] who exhibits some counterexamples. The author investigates the general case and proves that a generic point of a \(C^r\)-manifold endowed with a generic Riemannian structure has a unique antipode (i.e., farthest point). Furthermore, in the case of \(2\)-dimensional manifolds, such a point is joined to its antipode by at most three minimizing geodesics (\(r\geq2\)). Analogous results in the case of a convex surface are proved by \textit{T. Zamfirescu} [Proc. Am. Math. Soc. 123, No. 11, 3513--3518 (1995; Zbl 0851.52003)].
scientific article; zbMATH DE number 2101130