Let $U(\mathcal{H})$ be the group of unitary operators on a Hilbert space with the norm topology. Let $H\subset U(\mathcal{H})$ be a closed subgroup. 
Under which condidions (on the subgroup) is there a local cross section for the projection $p:U(\mathcal{H})\to U(\mathcal{H})/H$. 



Local cross  sections  for Unitary  group  in  a  hilbert space
You are basically asking: When is $p \colon U(\mathcal{H}) \to U(\mathcal{H})/H$ a principal $H$-bundle. Equipped with the norm topology $U(\mathcal{H})$ is a Banach-Lie group. There is a theorem for quotients of Banach-Lie groups by Glöckner and Neeb in a paper called [Banach-Lie Quotients, Enlargibility and Universal Complexifications](https://www.math.lsu.edu/~preprint/2001/gn2001.pdf). Remark II.5 in this paper might be helpful. 


To be precise they make the following definition:


* Let $G$ be a Banach-Lie group (over $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$), then a Banach Lie group $H \subset G$ is called an *analytic subgroup* if the inclusion map $\iota \colon H \to G$ is smooth and $\iota\_\* \colon Lie(H) \to Lie(G)$ is an embedding of topological Lie algebras.
* An analytic subgroup $H \subset G$ is called a *Lie subgroup* if $\iota \colon H \to G$ is a topological embedding.


These are quite natural conditions on subgroups. In Corollary II.4 they state: 



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The important bit is now that if you have a normal Lie subgroup $N$ of a Banach Lie group $G$, then $Lie(G/N) = Lie(G) / Lie(N)$. The quotient map $q \colon G \to G/N$ induces a map of Banach spaces $Lie(G) \to Lie(G) / Lie(N)$, which has a continuous (but not necessarily linear!) section by Michael's selection theorem (sometimes called the Bartle-Graves theorem in this context, if I remember correctly). Together with the exponential map, this implies that $G \to G/N$ has local sections.


