Non-existence of proper η-Einstein Kenmotsu manifolds

In 1972, Kenmotsu [A class of almost contact Riemannian manifolds, Tohoku Math. J.(2) 24 (1972) 93–103] defined the notion of Kenmotsu manifolds. He has also studied the properties of proper [Formula: see text]-Einstein Kenmotsu manifolds, and proved some basic interesting results. These results have been used by a plenty of authors in establishing their main results on Kenmotsu manifolds. Naturally, the question of existence of proper [Formula: see text]-Einstein Kenmotsu manifold is a prime question in the study of contact geometry. The main goal of this paper is to discuss the existence of proper [Formula: see text]-Einstein Kenmotsu manifolds and to refine some of Kenmotsu’s results. The results of this paper may be used as an erratum of many published papers on Kenmotsu manifolds. In this sequence, we also address the existence of conharmonically flat Kenmotsu manifolds, conformal Ricci flow and almost Ricci solitons on Kenmotsu manifolds.