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I t is well known that the topology of various path spaces on a complete riemannian manifold M is closely related to the existence of various kinds of geodesics on M. Classical Morse theory and the theory of closed geodesics are beautiful examples of this sort. The motivation for the present paper is the study of geodesics satisfying a very general boundary condition of which the above examples and the example of isometryinvariant geodesics are particular cases. In particular, we generalize a result of SullivanVigu6 [16]. Let 2 2 c M x M be a submanifold of the riemannian product M x M. An N-geodesic on M is a geodesic c: [0, 1]-~M which satisfies the boundary condition

The rational homotopy theory of certain path spaces with applications to geodesics