Certain homotopies in the space of closed curves

It is shown that a smooth homotopy of a Riemannian manifold induces a smooth homotopy of the space of closed curves, and that it is possible to pass to a parametrization of the curves that is proportional to the arc length by means of a certain homotopy in this space. Applications are given to the homology of the space of nonoriented closed curves on a sphere, and errors in some previous articles on this topic are corrected. Despite these errors, it turns out to be possible to repair the proofs of theorems of Klingenberg and Al'ber on closed nonselfintersecting geodesics on a sphere with a Riemannian metric satisfying the $1/4$-pinching condition on the curvature (and, in the Al'ber theorem, also the Morse condition).
Bibliography: 10 titles.