TITLE: Question on Hilbert Manifolds
QUESTION [9 upvotes]: I have a very basic question on Hilbert manifolds.
Consider the Hilbert space 
$$
\mathcal{H}:= L^2(S^1)
$$
with $S^1$ the unit circle. 
On $\mathcal{H}$ let us introduce the equivalence relation 
$$
f\sim g : \Leftrightarrow f(\cdot ) = g(\cdot + \alpha)\quad
\mbox{for some  }\alpha \in S^1.
$$
Now define the factor space 
$$
\overline{\mathcal{H}}:= \mathcal{H}/\sim.
$$
What is the structure of $\overline{\mathcal{H}}$? Is it a Hilbert manifold? If so, how to construct the smooth structure? 
I am particularly interested in computing a (Riemannian) distance between two elements of $\overline{\mathcal{H}}$.

REPLY [3 votes]: It probably is NOT a smooth manifold. I think finding a chart around the point corresponding to constants, namely, the fixed points of the action of the group of rotation, is problematic.More precisely, at a fixed point, there is not a well-defined tangent space.