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
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