LS-category and topological complexity of real torus manifolds and Dold manifolds of real torus type

The real torus manifolds are a generalization of small covers, and the Dold manifolds of real torus type are a class of non-trivial fibre bundles over the projective product spaces with real torus manifolds as fibres. In this paper, first, we compute the LS-category of these two types of manifolds and obtain sharp bounds on their topological complexities. We show that under certain hypotheses, the topological complexities of real torus manifolds of dimension $n$ are either $2n$ or $2n+1$.We figure out tight bounds for the topological complexity of generalized real Bott manifolds, and in many cases, the difference between these upper and lower bounds is less than 5. We compute the $\mathbb{Z}_2$-equivariant LS-category of small covers when the $\mathbb{Z}_2$-fixed points are path connected. In the end, we study the symmetric topological complexity of the above-mentioned manifolds and obtain exact values for infinitely many cases.