8 Takashi SHIOYA

intervals [2;,&i] (where i e (7z(a)]) satisfying [ai,bi] ^ (ai+ubi+i) and a(ai) =

&(bi) = pi for all i e \n(a)] such that p is a double point of a when and only

when p = pi for some i in [rc(a)]. The semi-regular arc a will be called a seraz-

regular curve when the closed interval on which it is defined is equal to R. Notice that

any closed subarc of a semi-regular arc is a semi-regular arc.

1.5. Classification of loops and biangles. In order to describe semi-

regular curves we now classify loops and biangles up to diffeomorphisms.

Recall that a loop of M is the image of a transversal immersion a of some

closed interval [a,b] with only one double point p = a(a) = a(b). Since M is

homeomorphic to R2 any loop a of M bounds a compact disk A(a) C M. In the

absence of a Riemannian structure on M one can not measure the inner angle 6 of

A(a) at p\ nevertheless (for any Riemannian structure on M) by transversality one

may claim that 0 * 0, 6 * n and 6 * 2n and decide (independently of the given

Riemannian structure) whether (L\) 6 e (0,;r), in which case A(a) is said to be a

teardrop or (L2) 6 e (n,27t), in which case A{a) is said to be a tomato.

teardrop tomato

Recall also that a biangle of M is the union of the images of two differentiable

embeddings a: [a\a] — M and /?: [b,b'] — M intersecting each other

transversely only at the two distinct points p = a(a) = /3(b) and p' = cc(a') -

/?(£'). Any biangle a u p of M bounds a compact disk D - A(a u j3) c