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Immersed surfaces and pencils of planes in 3-space

Published online by Cambridge University Press:  18 May 2009

F. J.Craveiro de Carvalho
F. J.Craveiro de Carvalho
Affiliation:
Departamento de Matemática, Universidade de Coimbra, Portugal
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Let M be a compact connected boundaryless surface and f: M → ℝ3 a smooth immersion transverse to a straight line L. Thus there is an even number p of points xεM such that f(x)εL. Under further transversality assumptions on f (see §3) there is a finite number q of points x of M such that the plane containing f(x) and L touches f(M) at f(x). These assumptions are mild in the sense that they hold for any f in an open dense subset of the space of smooth immersions under consideration. Suppose that the Gaussian curvature of f(M) is positive at q+ of these points and negative at q−, with q = q++ q−. Then

where e(M) denotes the Euler number of M.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 22 , Issue 2 , July 1981 , pp. 133 - 136
Copyright
Copyright © Glasgow Mathematical Journal Trust 1981

References

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