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1996•Experimental MathematicsRequires access

On a Conjecture of Carathéodoryr Analyticity Versus Smoothness

Carlos Gutiérrez, Francesco Mercuri, Federico Sánchez-Bringas

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Abstract

We show that, under mild nonflatness conditions, for any r ≥ 3 and any C r -immersion of a surface into R3 with an isolated urnblhc point there exist an analytic surface with an isolated umbilic of the same index. The connection of this with Caratheodory's Conjecture on umbilics is discussed.

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What this paper is about

We show that, under mild nonflatness conditions, for any r ≥ 3 and any C r -immersion of a surface into R3 with an isolated urnblhc point there exist an analytic surface with an isolated umbilic of the same index. The connection of this with Caratheodory's Conjecture on umbilics is discussed.

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OpenAlex reports 10 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

We show that, under mild nonflatness conditions, for any r ≥ 3 and any C r -immersion of a surface into R3 with an isolated urnblhc point there exist an analytic surface with an isolated umbilic of the same index. The connection of this with Caratheodory's Conjecture on umbilics is discussed.

Key concepts: Mathematics, Conjecture, Smoothness, Connection (principal bundle), Immersion (mathematics), Pure mathematics, Surface (topology), Mathematical analysis

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