Let \((M, \alpha)\) be a contact manifold. The contact form \(\alpha\) is said to be \(K\)-contact if there exists a contact metric \(g\) which is invariant under the characteristic vector field \(v\) of \(\alpha\), i.e. \({\mathcal L}_v g= 0\). The author writes that there seems to be a confusion in the literature whether or not the requirement on \(g\) to be contact is necessary in the definition of \(K\)-contact form, and proves a theorem which implies that it is not. Namely, if the characteristic of \(\alpha\) is Riemannian, then \(\alpha\) is a \(K\)-contact form. The author also finds sufficient conditions under which a \((2n+ 1)\)-dimensional Riemannian manifold \((M, g)\) admits a \(K\)-contact form. He proves that if there exists a unit Killing vector field \(w\) such that each 2-plane \(\sigma_x\), \(x\in M\), with \(w(x)\in \sigma_x\), has positive sectional curvature, then there exists a \(K\)-contact form \(\alpha\) on \(M\) with characteristic vector field \(w\). This proposition generalizes the result in \textit{Y. Hatakeyama}, \textit{Y. Ogawa} and \textit{S. Tanno} [Tôhoku Math. J., II. Ser. 15, 42-48 (1963; Zbl 0196.54902)].