# The Tangent Bundle of the Space of CR Structures on S^(2n+1)

Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for quantities determined by a choice of Riemannian metric on $M$ . . . perhaps $\lambda_1(\int_M \omega_g)^{2/n}$, where $\lambda_1$ is the first nonzero eigenvalue of a Laplace-type operator, or maybe the zeta-regularized determinant of the conformal Laplacian.

The setting for this variational problem is the space of Riemannian metrics $\mathcal{M}$ on $M$. This is a tame Frechet manifold and is an open convex cone inside the tame Frechet space $\Gamma^\infty\!(S^2T^\ast\!M)$ of symmetric covariant 2-tensors on $M$ (Hamilton, 1982). These objects are standard in conformal geometry. In particular, if $M=S^{n}$, then $\Gamma^\infty\!(S^2T^\ast\!M)$ is the space of sections of a vector bundle associated to a representation of the conformal group of $M$, and variational problems can be addressed via the action of the conformal group (Møller & Ørsted, 2009).

My concern is whether there is an analogue of this picture for strongly pseudoconvex CR manifolds, in particular for odd-dimensional spheres. What is the tangent bundle of the space of strongly pseudoconvex CR structures? What is the tangent bundle of the space of all CR structures? Is the space of strongly pseudoconvex CR structures on the sphere a tame Frechet manifold sitting as an open convex cone inside some tame Frechet space of sections of some vector bundle over $S^{2n+1}$? Is this vector bundle associated to a representation of a Lie group?