The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.