In [Algebras Groups Geom. 7, 145-152 (1990;

Zbl 0782.53025)] the author proved the following: I) On a $K$-contact manifold a second order symmetric parallel tensor field is a constant multiple of the metric. II) On a Sasakian manifold there are no nonzero parallel 2-forms. In the present paper the author proves a theorem which contains each of the above as a special case. Let $M$ be a contact metric manifold and let $\xi$ denote the characteristic vector field of the contact structure. If the $\xi$-sectional curvature, $K(\xi,X)$, is nowhere vanishing and independent of the direction of $X$, then a second order parallel tensor field on $M$ is a constant multiple of the metric tensor. Examples of non-$K$-contact, contact metric manifolds satisfying the condition may be found in the reviewer’s paper with {\it H. Chen} [Bull. Inst. Math., Acad. Sin. 20, No. 4, 379-383 (1992;

Zbl 0767.53023)]. In addition to I) and II) being consequences of this result, one also has the following theorem of {\it S. Tanno} [Proc. Japan Acad., Ser. A 43, 581-583 (1967;

Zbl 0155.498)] as a corollary: If the Ricci tensor field is parallel on a $K$-contact manifold, then it is an Einstein space.