Immersed surfaces and pencils of planes in 3-space

Extract

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.