  A topological group is constructed which is homotopy equivalent to the
pointed loop space of a path-connected Riemannian manifold $M$ and which is
given in terms of "composable small geodesics" on $M$. This model is analogous
to J. Milnor's free group construction \cite{Milnor} which provides a model for
the pointed loop space of a connected simplicial complex. Related function
spaces are constructed from "composable small geodesics" which provide models
for the free loop space of $M$ as well as the space of continuous maps from a
surface to $M$.
