Gutierrez, Carlos; Mercuri, Francesco; Sánchez-Bringas, Federico On a conjecture of Carathéodory: Analyticity versus smoothness. (English) Zbl 0862.57023 Exp. Math. 5, No. 1, 33-37 (1996). 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. Cited in 1 ReviewCited in 3 Documents MSC: 57R42 Immersions in differential topology Keywords:immersion; surface; umbilic point; analytic surface PDFBibTeX XMLCite \textit{C. Gutierrez} et al., Exp. Math. 5, No. 1, 33--37 (1996; Zbl 0862.57023) Full Text: DOI EuDML EMIS References: [1] Asperti A. C., Boletim Soc. Bras. Mat. 11 pp 191– (1980) · Zbl 0573.53029 · doi:10.1007/BF02584637 [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 · doi:10.1007/BF01174209 [5] Feldman E. A., Trans. Amer. Math. Soc. 127 pp 1– (1967) · doi:10.1090/S0002-9947-1967-0206974-1 [6] DOI: 10.1016/0022-0396(77)90136-X · Zbl 0346.58002 · doi:10.1016/0022-0396(77)90136-X [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) · doi:10.1007/BFb0061423 [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 · doi:10.2307/1968821 [13] Klotz T., Comm. Pure Appl. Math. 12 pp 277– (1959) · Zbl 0091.34301 · doi:10.1002/cpa.3160120207 [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 · doi:10.1007/BF01120328 [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 · doi:10.1112/blms/24.2.176 [18] Titus C. J., Acta Math. 131 pp 43– (1973) · Zbl 0301.53001 · doi:10.1007/BF02392036 [19] Yau S. T., Seminar on Differential Geometry (1982) · Zbl 0471.00020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.