Let \(T\) be a torus. The Chang--Skjelbred Lemma says that the rational cohomology of an equivariantly formal \(T\)-manifold \(M\) is determined by the topology of the one-skeleton of the \(T\)-action on \(M\), i.e. by the topology of the union of all orbits of dimension at most one.

In these talks I will report on two projects (one joint with L. Kennard and B. Wilking, the other joint with O. Goertsches) in which we apply this lemma in the case that \(M\) is positively or non-negatively curved. In these projects we show that, under certain assumptions on the action, the rational cohomology \(M\) is isomorphic to the cohomology of some standard model spaces