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