Consider the odd-dimensional sphere $S^{2n+1} \subset \mathbb{C}^{n+1}$. One may talk variously about its structure as a contact, CR or Einstein-Sasaki manifold, but I'm looking for some specific down-to-earth detail that is hard to track down, namely charts that are complements of points and which are 'unitary' in the following sense.
The tangent space to $S^{n+1}$ at a point $x$ has a subspace isomorphic to the hermitian complement to the complex span of $x$. These patch together to give a complex vector bundle on the sphere -- and I'm fairly sure this is the holomorphic tangent bundle $HS^{2n+1}$ in the CR setting. Taking for $U$ the standard complement of the south pole, there is an isomorphism (even a CR isomorphism!) $\phi\colon \mathbb{C}^n\times \mathbb{R}\simeq U$$\phi\colon Heis_n\simeq U$, where $Heis_n$ is the Heisenberg group, diffeomorphic to $\mathbb{C}^n\times\mathbb{R}$. Hence we can consider the induced isomorphism $T(\mathbb{C}^n)\times \mathbb{R} \to HU$$H\, Heis_n \to HU$ and ask whether it is unitary with respect to the standard hermitian structure on the left, and the one induced from $\mathbb{C}^{n+1}$ on the right. This is all controlled by the isomorphism $\phi$, and if I'm not mistaken in my calculations, taking standard stereographic projection rewritten in complex coordinates doesn't do the trick (which is, making the relevant adjustments, only orthogonal).
What's an explicit chart for an odd-dimensional sphere (considered as embedded in $\mathbb{C}^{n+1}$) satisfying the above condition?
Asking for such a chart (and analogously over the complement of the north pole) is equivalent to giving a local section of $SU(n+1) \to S^{2n+1}$ over $U$ (and $V$). Note that from a Riemannian point of view I want to consider the sphere with the homogeneous metric from the isomorphism $S^{2n+1} \simeq SU(n+1)/SU(n)$.
I have a hard time believing no one has written this down before.