Let \((M,g)\) be a compact orientable Riemannian manifold and \((T^1M,g_s)\) the unit tangent sphere bundle of \(M\) equipped with the Sasaki metric. A unit vector field \(V\) on \(M\) determines a map between \((M,g)\) and \((T^1M,g_s)\). The vector field \(V\) is said to be harmonic if it is a critical point of the corresponding energy functional. Now assume that \((M,g)\) is equipped with a contact metric structure. Then the corresponding characteristic vector field \(\xi\) on \(M\) is a unit vector field. If \(\xi\) is harmonic then \(M\) is said to be an \(H\)-contact manifold. The main result of the author says that a contact metric manifold \((M,g)\) is an \(H\)-contact manifold if and only if the characteristic vector field \(\xi\) is an eigenvector of the Ricci tensor of \((M,g)\) everywhere. For dimension \(3\) this was already proved by \textit{J. C. González-Dávila} and \textit{L. Vanhecke} in [J. Geom. 72, 65--76 (2001; Zbl 1005.53039)]. The author then investigates the relation between \(H\)-contact manifolds and some known classes of contact metric manifolds. He also investigates the topology of \(H\)-contact manifolds.