Compression theorems for surfaces and their applications

Let M be a complete glued surface whose sectional curvature is greater than or equal to k and \triangle pqr a geodesic triangle domain with vertices p, q, r in M . We prove a compression theorem that there exists a distance nonincreasing map from \triangle pqr onto the comparison triangle domain \widetilde \triangle pqr in the two-dimensional space form with sectional curvature k . Using the theorem, we also have some compression theorems and an application to a circular billiard ball problem on a surface.