Contact transformations, Vector fields, Contact manifolds
First we improve a result of Tanno that says "If a conformal vector field on a contact metric manifold M is a strictly infinitesimal contact transformation, then it is an infinitesimal automorphism of M" by waiving the "strictness" in the hypothesis. Next, we prove that a (k, μ)-contact manifold admitting a non-Killing conformal vector field is either Sasakian or has k = –n – 1, μ = 1 in dimension > 3; and Sasakian or flat in dimension 3. In particular, we show that (i) among all compact simply connected (k, μ)-contact manifolds of dimension > 3, only the unit sphere S2n+1 admits a non-Killing conformal vector field, and (ii) a conformal vector field on the unit tangent bundle of a space-form of dimension > 2 is necessarily Killing.
Sharma, Ramesh and Vrancken, Luc, "Conformal Classification of (k, μ)-Contact Manifolds" (2010). Mathematics Faculty Publications. 6.
Ramesh Sharma and Luc Vrancken. Conformal classification of (k,mu)-contact manifolds, Kodai Mathematical Journal 33 (2010), 267-282.