Tohoku Mathematical Journal
2000
June
SECOND SERIES VOL. 52, NO. 2

Tohoku Math. J.
52 (2000), 163-172

Title LINES OF PRINCIPAL CURVATURE AROUND UMBILICS AND WHITNEY UMBRELLAS

Author Ronaldo Garcia, Carlos Gutierrez and Jorge Sotomayor

(Received February 26, 1998, revised November 4, 1999)
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-Poincare characteristic in terms of the number of umbilics and umbrellas.

1991 Mathematics Subject Classification. Primary 58F14; Secondary 53C03, 53A05, 58F10, 58F18.
Key words and phrases. Whitney umbrella singularity, lines of curvature, principal configuration.

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