ABSTRACT
Investigation of the properties of geodesics on closed orientable surfaces of genus one has been made in certain special cases. A study of the geodesics on a more general type of closed surface of revolution of genus one, including the torus as a special case, has been made by Kimball. Assuming suitable conditions concerning the class of the functions defining the homeomorphism with the torus, the problem of studying the geodesics on the given surface becomes that of determining the geodesics on a two-dimensional Eiemannian manifold with periodic coefficients, and can be conveniently considered in a plane. Moreover these-geodesics can be so chosen that the corresponding constants are uniformly bounded. The existence of the geodesics of periodic type is known, so that what is essentially new is the existence of those of non-periodic type.
