Behavior of distant maximal geodesies in 2-dimensional manifolds 9

M. Independently of any given Riemannian structure on M one of the three

following possibilities holds. (Bj) The two inner angles of D are in (0,zr), in which

case D is said to be a lemon. (B2) One inner angle of D is in (0,;r) and the other

in

(TT,2TZ:),

in which case D is called a heart. (B3) Both angles are in {n,2it), in

which case D is defined to be an apple.

lemon heart apple

1.6. Description of semi-regular curves in terms of loops and

biangles. Let a be a nonsimple semi-regular curve i.e. a semi-regular curve such

that n(a) 1, (a condition which means that the set {pi}[e[n(a)) is nonempty). Since

P\ Pi ••• the arc ct([a\,b\\) is a loop bounding a disk B\, which must be a

teardrop, and for each integer i such that 2 i ~n(a) the union of subarcs

a([a^uai\) and a([&M,&/]) is a biangle (which does not intersect the loop and

the other biangles except of course /?,_i and/?/) bounding a disk B\ which satisfies

one of the two sets of equivalent conditions.

(i) sgn(p;_i) * sgn(p/) = the setB[- {p\-\\ does not intersect U Bj = the

disk Bi is a lemon.

(ii) sgn(p/_i) = sgn(pi) = the set Int(5z) U {p/_i} contains U By = the disk

5f- is a heart with its angle greater than 7rat/?;_i.

This is indeed the case because since a is proper, a tomato or an apple (as well as