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 $\overline{\mathcal{H}}$.