  In this paper, we compute the LS-category and equivariant LS-category of a
small cover and its real moment angle manifold. We calculate a tight lower
bound for the topological complexity of many small covers over a product of
simplices. Then we compute symmetric topological complexity of several small
covers over a product of simplices. We calculate the LS one-category of real
Bott manifolds and infinitely many small covers.
