Behavior of distant maximal geodesies in 2-dimensional manifolds 7

the two following situations may occur.

(i) [a\,b\] n [02^2] = 0»

*n

which case p\ and pi are said to be

independent.

(ii) [ai,b\] n [02,^2] is a nonempty interval / * [2i,Z?;] for / = 1,2, in

which case/?i and/?2 are said to /m&.

P\ and^2 mz independent. p\ and pi link.

1.4. Definition. A proper transversal immersion of some (not necessarily

compact) closed interval of R into M is called a semi-regular arc when the set of

double points of a is totally ordered by the previously defined order relation.

A semi-regular arc

In other words, setting ~n{a) := n+(cc) + n_(a) and for all n e N U {00}

denoting by [n] the set of all integers / such that 1 i n, the proper transversal

immersion a is said to be semi-regular when there exists a sequence of closed