Consider a \((2n+1)\)-dimensional contact metric manifold \((M,\alpha, g)\). It is a \(K\)-contact manifold if \(g\) is preserved by the Reeb flow of \(\alpha\) and if there exists an almost complex structure \(J\) on \(ker\;\alpha\) such that \(g(X,Y)=d\alpha(X,JY)\) where \(X,Y\) are sections in \(ker\;\alpha\). The \(K\)-contact manifolds have been studied intensively in the theory of almost contact manifolds (especially in the theory of Sasakian manifolds). The authors study the torus actions on \(K\)-contact manifolds. The torus \(T\) is obtained by the closure of the Reeb flow of \(\alpha\) in the isometry group of \((M,g)\).
The following results are proved or are studied from the point of view of their equivalence with some other results from the theory of \(K\) contact manifolds. Theorem: The \(T\)-action on \(M\) is Cohen-Macaulay. Theorem: The \(\mathfrak{\alpha}\)-action on \((M,\mathcal{F})\) is equivariantly formal. (The set \(\mathcal{F}\) is the orbit foliation of the Reeb flow.) Some other results are obtained, concerning the basic cohomology of \((M,\mathcal{F})\). Theorem: The Reeb flow of \(\alpha\) has at least \(n+1\) closed orbits. Theorem: If the closed Reeb orbits of \(\alpha\) are isolated, then their number is exactly \(n+1\) if and only if \(M\) is a real cohomology sphere.
Finally, the authors use a version of GKM theory in order to compute the equivariant cohomology \(H^*_T(M)\) as a graded \(S(t^*)\) algebra.