## Tohoku Mathematical Journal

### Lines of principal curvature around umbilics and Whitney umbrellas

#### Abstract

In this paper is studied the configuration of lines of curvature near a Whitney umbrella which is the unique stable singularity for maps of surfaces into $R^3$. The pattern of such configuration is established and characterized in terms of the 3-jet of the map. The result is used to establish an expression for the Euler-Poincaré characteristic in terms of the number of umbilics and umbrellas.

#### Article information

Source
Tohoku Math. J. (2), Volume 52, Number 2 (2000), 163-172.

Dates
First available in Project Euclid: 3 May 2007

https://projecteuclid.org/euclid.tmj/1178224605

Digital Object Identifier
doi:10.2748/tmj/1178224605

Mathematical Reviews number (MathSciNet)
MR1756092

Zentralblatt MATH identifier
0973.37012

Subjects
Primary: 58K15: Topological properties of mappings
Secondary: 53A05: Surfaces in Euclidean space

#### Citation

Garcia, Ronaldo; Gutierrez, Carlos; Sotomayor, Jorge. Lines of principal curvature around umbilics and Whitney umbrellas. Tohoku Math. J. (2) 52 (2000), no. 2, 163--172. doi:10.2748/tmj/1178224605. https://projecteuclid.org/euclid.tmj/1178224605

#### References

• [B-F] J W. Bruce and D L Fidal, On Binary Differential Equation and Umbilics, Proc Roy Soc Edinburgh, Sect A 111 (1989), 147-168.
• [Dar] G Darboux, Sur la forme des lignes de courbure dans la voisinage d'un ombilic. Leons sur la Theorie de Surfaces, vol IV, Note 07, Gauthiers Villars, Paris, 1896.
• [Gui] V Gunez, Positive quadratic differential forms and foliations with singularities on surfaces, Trans Ame Math. Soc 309, 2 (1988), 375^13
• [G-G] M Golubitsky and V Guillemin, Stable Mappings and their Singularities, Grad Texts in Math 14, Springe Verlag, New York, 1973
• [GS1] C. Gutierrez and J. Sotomayor, Structural Stable Configurationsof Lines of Principal Curvature, Asterisqu 98-99(1982), 195-215
• [GS2] C Gutierrez and J. Sotomayor, Lines of principal curvature for mappings with Whitney umbrella singulari ties, Thoku Math. J 38 (1986), 551-559
• [GS3] C. Gutierrez and J Sotomayor, Lines of Curvature and Umbilical Points, IMPA, Rio de Janeiro, 199
• [Hop] H. Hopf, Differential Geometry in the Large, Lecture Notes in Math 1000, Springer Verlag, 197
• [I-M] S Izumiya and W Marar, On topologically stable singular surfaces in a 3-manifold, J of Geometry 5 (1995), 108-119
• [SGI] R Garcia and J Sotomayor, Lines of Curvature near Singular Points of Implicit Surfaces, Bull Sc Mat 117(1993), 313-331
• [SG2] R Garcia and J Sotomayor, Lines of Curvature on Algebraic Surfaces, Bull Sc Math. 120 (1996), 367 395
• [Spi] M Spivak, Introduction to Comprehensive Differential Geometry, Vol. III, Publish or Perish Inc., Berkeley, 1980
• [Whi] H. Whitney, The general type of singularity of a set of 2n -- 1 smooth functions of n variables, Duke Math J. 10(1943), 161-172
• [Wes] J West, The Differential Geometry of the Crosscap, Ph D Thesis, University of Liverpool, 1995, 1-17