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

*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

*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.