We are interested in the basic question as to whether a nowhere zero vector field on a smooth compact manifold, with or without boundary, has a periodic orbit. For example, forty years ago Smale asked whether every nonvanishing smooth vector field on the solid torus had a periodic orbit. In 1995, K. Kuperberg answered this and the more famous Seifert Conjecture in the negative. Nonetheless, in this talk we present a series of positive results on the existence of periodic orbits that are valid, under some fairly natural hypotheses, in much greater generality than the setting of Smale's original question. In particular, we present general results on the existence of periodic orbits for vector fields on n-manifolds having an ``angular'' one-form. Roughly speaking, an angular one-form is a closed nonsingular one-form which is a generalized form of angular velocity analogous to the common interpretation of a Lyapunov function as a generalized form of energy.
In the spirit of Thurston's Geometrization Program, we illustrate these results in the case of 3-dimensional manifolds. Moreover, using the validity of the Poincare Hypothesis in all dimensions we prove a general converse asserting that any asymptotically stable periodic orbit is contained in a positively invariant solid n-torus on which there exists an angular one-form.