THE TOPOLOGICAL AND DYNAMICAL PROPERTIES OF NONORIENTABLE SURFACES

Four topological and dynamical properties of nonorientable surfaces are proved. The first is that for every continnous flow defined on any nonorientable closed surface, there exist periodic or singular closed orbits. In the case of the projective plane, it confirms a conjecture of professor Ye Yian-qian in his lecture notes "dynamicaL systems oil sxirfaces". Secondly, the author gives an exact upper bound of the number of closed curves on nonorientable surfaces, which do not intersect each other and the complement of their sum is still connected. The third is concerned with the upper and lower bound of the number of the periodic or singular closed orbits with certain properties. The last is related to the connectedness of the complement of a lifting curve on two-fold covering space. The first property may be considered as a generalization of Kneser theorem from Klein bottle to general nonorientable surfaces and the second as a generalization of 4] Theorem 9.3.6 from orientable surfaces to nonorientable surfaces.