Let $S^{2n-1}\subset\mathbb{C}^{n}$, and denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hermitian product. Then
$$
\mathcal{C}_p:=\{\xi\in T_pS^{2n-1}\mid\langle p,\xi\rangle=0\},\quad p\in S^{2n-1},
$$
defines a *contact structure* on $S^{2n-1}$, i.e., a 1-codimensional completely non-integrable distribution (in terms of contact forms, $\mathcal{C}$ may be thought of as the conformal class $[\theta]$ uniquely defined by $\mathcal{C}=\ker\theta$). I will call it **standard**.

QUESTION:How many contact structures $\mathcal{C}$ one can equip the sphere $S^{2n-1}$ with, in such a way that the corresponding Lie group $\mathrm{Cont}\,(S^{2n-1})$ of contactomorphisms contains a finite-dimensional Lie subgroup $G$ acting transitively on $S^{2n-1}$?REMARK: By "how many" I mean, of course, up to equivalence via diffeomorphisms. Moreover, if this will facilitate the answer, some extra topological (e.g., compactness, simply-connectedness) and/or algebraic (e.g.,semi-simplicity) property can be added to $G$.

Just to motivate the question, observe that, if $\mathcal{C}$ is the above standard structure, then $G$ can be taken as the unitary group $\mathrm{U}(n)$, so that there is - at least - the equivalence class of the standard contact structure. I'd like to know whether there are others.

**SIDE QUESTIONS.** Even without an answer to the main question, perhaps some clues/references concerning the topics below will help me:

- how many contact structures there are on odd-dimensional spheres?
- in how many ways one can construct spheres as homogeneous spaces?

BELOW THERE IS A

LONG EDITFOLLOWING R. BRYANT'S LAST REMARK.

According to R. Bryant's remark, there is a unique "up to equivalence" homogeneous contact structure on the odd-dimensional sphere. I'm trying now to understand why.

First, I lack the notion of "up to equivalence" in the context of homogeneous spaces. Here it goes my own intuition.

Let $M$ be a smooth manifold which is homogeneous w.r.t. two (in principle) different Lie groups $G$ and $\widetilde{G}$, i.e., $$M=\frac{G}{H}=\frac{\widetilde{G}}{\widetilde{H}}.$$

DEFINITION. The two structures of homogeneous manifolds are

equivalentiff there exists $\phi\in\textrm{Hom}\,(G,\widetilde{G})$ such that: 1) $\phi(\widetilde{H})\subseteq \widetilde{H}$ and 2) $[\phi]\in\textrm{Diff}\, M$, where $\frac{G}{H}\stackrel{[\phi]}{\longrightarrow}\frac{\widetilde{G}}{\widetilde{H}}$ is the induced map.

SIDE QUESTION:is this definition correct? does it have some relevant application? does it fit in some category-theoretic approach to homogeneous spaces?

Second, assuming that the above definition is the correct one, how to prove that $S^{2n-1}=\frac{SU(n)}{SU(n-1)}$ is the unique homogenous manifold structure on $S^{2n-1}$?

So, given another structure $S^{2n-1}=\frac{G}{H}$, all boils down to prove that there is a group homomorphism $\phi:G\to SU(n)$, such that $\phi(H)\subseteq SU(n-1)$ and $[\phi]$ is a diffeomorphism.

SIDE QUESTION:is this the right way to tackle with the problem? can infinitesimal arguments be used instead? is there any book/paper where this sort of problems are dealt with?

Finally, going back to the key topic of this post, suppose that $(M,\mathcal{C})$ and $(M,\widetilde{\mathcal{C}})$ are two different homogeneous contact structure on the same manifold $$ M=\frac{G}{H}=\frac{\widetilde{G}}{\widetilde{H}}, $$ i.e., $\mathcal{C}$ is $G$-invariant and $\widetilde{\mathcal{C}}$ is $\widetilde{G}$-invariant.

Infinitesimally, this means that $$ \mathfrak{g}=\mathcal{C}_o\oplus\mathbb{R}Z \oplus\mathfrak{h};\quad \widetilde{\mathfrak{g}}=\widetilde{\mathcal{C}}_o\oplus\mathbb{R}\widetilde{Z} \oplus\widetilde{\mathfrak{h}} $$ where $Z$ and $\widetilde{Z}$ are the Reeb vector fields, and $$ \mathcal{C}_o\oplus\mathbb{R}Z = \widetilde{\mathcal{C}}_o\oplus\mathbb{R}\widetilde{Z} = T_oM $$ is the tangent space at the origin. Since there a linear transformation $h\in\mathrm{Aut}\, T_oM$ such that $h(\mathcal{C}_o)=\widetilde{\mathcal{C}}_o$, the local diffeomorphism $$ \widehat{h}:=\widetilde{\exp}\,\circ h\circ \exp^{-1} $$ sends $\mathcal{C}$ to $\widetilde{\mathcal{C}}$.

SIDE QUESTION:can this $\widehat{h}$ be used to prove Bryant's claim on the uniqueness of the homogeneous contact structure on $S^{2n-1}$? more generally, is there a criterion to patch together these diffeomorphisms and obtain a global diffeomorphism, thus proving the uniqueness of homogeneous contact structure onanyhomogeneous manifold? if not, are there examples of non-equivalent homogeneous contact structures on the same manifold?

Contact homogeneous spaces, Funktsional. Anal. i Prilozhen. 24 (1990), no. 4, 74--75. English translation in Funct. Anal. Appl. 24 (1990), no. 4, 324–325 (1991). See the MathSciNet entry. $\endgroup$