Theorem A of Chapter I states that a periodic flow on a Riemannian manifold with each trajectory geodesic is equivalent to a circle action with the same orbits. Using a similar method of proof we obtain a theorem on pointwise periodic hhomeomorphisms of immersed submanifolds. This generalises a result of N. Weaver. As an application, we show that if M is a two-dimensional Riemannian manifold with all closed geodesics then the geodesic loops of M are all of equal length. In Chapter II, our main theorem asserts that a foliated Riemannian manifold which is foliated by totally geodesic compact leaves has finite holonomy. This result has some application to isometric immersions of Riemannian manifolds in spaces of constant curvature.