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!) $\phi\colon \mathbb{C}^n\times \mathbb{R}\simeq  U$. Hence we can consider the induced isomorphism $T(\mathbb{C}^n)\times \mathbb{R} \to HU$ 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.