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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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References:

[1] Asperti A. C., Boletim Soc. Bras. Mat. 11 pp 191– (1980) · Zbl 0573.53029
[2] Blaschke W., Vorlesungen über Differentialgeometrie, III: Differentialgeometrie der Kreise und Kugeln (1929) · JFM 55.0422.01
[3] Bonnet O., J. Liouville (Sér. 2) 5 pp 153– (1860)
[4] Bol G., Math Z. 49 pp 389– (1943) · Zbl 0028.42501
[5] Feldman E. A., Trans. Amer. Math. Soc. 127 pp 1– (1967)
[6] DOI: 10.1016/0022-0396(77)90136-X · Zbl 0346.58002
[7] Darboux G., Le\c{}ons sur la Theorie des Surfaces (1896)
[8] Guillemin V., Differential Topology (1974)
[9] Gutierrez C., Geometric Dynamics pp 332– (1983)
[10] Gutierrez C., Bifurcation, Ergodic Theory and Applications pp 195– (1982)
[11] Gutierrez C., ”Lines of curvature and umbilic points on surfaces” (1993)
[12] Hamburger H., Ann. of Math. 41 pp 63– (1940) · Zbl 0023.06902
[13] Klotz T., Comm. Pure Appl. Math. 12 pp 277– (1959) · Zbl 0091.34301
[14] Lang M., Dissertation, in: ”Nabelpunkte, Krümmungslinien, Brennflächen und ihre Metamorphosen” (1990) · Zbl 0722.53002
[15] Ramírez-Galarza A. I., Ann. Global Anal. Geom. 13 pp 129– (1995) · Zbl 0836.53003
[16] Scherbel H., ”A new proof of Hamburger’s Index Theorem on umbilical points”
[17] Smyth B., Bull. Lon. Math. Soc. 24 pp 176– (1992) · Zbl 0763.53007
[18] Titus C. J., Acta Math. 131 pp 43– (1973) · Zbl 0301.53001
[19] Yau S. T., Seminar on Differential Geometry (1982) · Zbl 0471.00020
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