1. The semi-regular curves in a differentiable plane

In this paragraph we introduce preliminary notions needed in order to state the main

results of Chapter 2.

1.1. Proper transversal immersion. Let M be any smooth surface. A

differentiable mapping a of a (not necessarily compact) interval / of R into M is

said to be a weakly transversal immersion (resp. a transversal immersion) if it

satisfies Conditions (i) and (ii) (resp. Conditions (i), (ii) and (iii)).

(i) (immersibility condition) 6c(t) := &- a(t) * 0 for all t e /.

at

(ii) (source transversality condition) whenever a(a) = a(b) = p for a * b,

the tangent vectors d(a) and d(b) are linearly independent in TpM.

(iii) (target transversality requirement) The mapping a has no triple points, in

other words there exist no a, b, c e / such that a * b * c * a and that a(a)

= a(b) = a(c).

1.1.1. Lemma. Let a be a proper transversal immersion of a not necessarily

compact interval / into a smooth surface M. Then the set of double points of a is a

discrete subset of M.

•

Suppose from now on that the surface M is diffeomorphic to

R2.

Assume for

convenience that M is oriented and suppose that a b. A crossing point p =

a(a) = a(b) of a will be said to have a positive sign ( sgn(p) = 1 ) when the

basis (d(a\d(b)) has positive orientation and a negative sign ( sgn(p) = -1 )

otherwise.

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