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A note on Ogiue-Takagi conjecture on a characterization of Euclidean 2- spheres. (English) Zbl 0763.53005

The authors’ main result is the following theorem: A simply connected complete \(C^ 2\) surface in \(E^ 3\) which contains two transversal circles through each point must be a plane or a sphere. This gives a partial answer to a conjecture presented and discussed in a paper by K. Ogiue and R. Takagi [Tsukuba J. Math. 8, 171-182 (1984; Zbl 0544.53002)]. It has been shown by the second author [J. Geom. 24, 123-130 (1985; Zbl 0571.53001)] that a complete simply connected smooth surface in \(E^ 3\) is a plane or a sphere if through each point there pass three circles.

MSC:

53A05 Surfaces in Euclidean and related spaces
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