By the Poincaré–Hopf theorem, every ovaloid has at least one umbilic. In this paper we extend this result to the more general case of complete positively curved surfaces in whose shape operator satisfies inf and sup .
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Francisco Fontenele, Frederico Xavier, Finding umbilics on open convex surfaces. Rev. Mat. Iberoam. 35 (2019), no. 7, pp. 2035–2052DOI 10.4171/RMI/1109