On $K$-contact metric structures

We obtain a complete system of structure equations of $K$-contact manifolds and study connections between various characteristics of isotropy of $K$-contact manifolds: the constancy of $\Phi$-sectional curvature, the axiom of $\Phi$-holomorphic planes, etc. We prove that a $K$-contact manifold is locally symmetric if and only if this manifold is conformally flat, and obtain a complete classification of such manifolds.