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On a conjecture of Carathéodory: Analyticity versus smoothness. (English) Zbl 0862.57023

Summary: We show that, under mild nonflatness conditions, for any \(r\geq 3\) and any \(C^r\)-immersion of a surface into \(\mathbb{R}^3\) with an isolated umbilic point there exist an analytic surface with an isolated umbilic of the same index. The connection of this with Carathéodory’s Conjecture on umbilics is discussed.

MSC:

57R42 Immersions in differential topology
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