I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g−1)$ pants bounded by $3$ geodesics. It seems reasonable to think the same can be done for a closed surface of genus $g>1$ carrying an arbitrary metric: i.e. to decompose it into $2(g−1)$ pieces which are all homeomorphic to a disc with two holes and bounded by geodesic. is any one have a good reference for that ? Thx

This question have been first posted on mathstack exchange, without any success...