Sasakian metric as a Ricci soliton and related results

We prove the following results: (i) a Sasakian metric as a non-trivial Ricci soliton is null ηη-Einstein, and expanding. Such a characterization permits us to identify the Sasakian metric on the Heisenberg group H2n+1H2n+1 as an explicit example of (non-trivial) Ricci soliton of such type. (ii) If an ηη-Einstein contact metric manifold MM has a vector field VV leaving the structure tensor and the scalar curvature invariant, then either VV is an infinitesimal automorphism, or MM is DD-homothetically fixed KK-contact.