Some problems concerning (partially) free maps and the sets of (partial) isometries

Free maps $f\colon M\to R^q$ are $C^2$ maps such that their first and second partial derivatives vectors are linearly independent at every point. For partially free maps it is sufficient that this is true for the partial derivatives along the directions of some fixed vector subbundle $H$ of $TM$. Similarly, partial immersions $f\colon M\to R^q$ are such that the pull-back through f of the euclidean metric is a Riemannian metric on some vector subbundle $H$ of $TM$.
By the Nash-Gromov theory of isometric embeddings there is a strict relation between (partially) free maps and the set of (partial) isometries.
We present here some recent results (joint work with G. D'Ambra and A. Loi) on the existence of isometric immersions when the dimension $q$ of the target space is so small that free maps cannot arise and on the existence of partial isometries in “critical dimension”.