Yes, the manifold question is completely answered [here][1]. Namely $\exp_n X^k$ (where $X^k$ is a $k$-manifold) is a manifold *if and only if* $k=1, n=3$ or $k=n=2.$ This is Theorem 1.3 in the referenced paper. **EDIT** Also, of course if $n=1,$ though the authors overlook this...


  [1]: http://arxiv.org/pdf/0902.3773v2