  We show that any effective isometric torus action of maximal rank on a
compact Riemannian manifold with positive (sectional) curvature and maximal
symmetry rank, that is, on a positively curved sphere, lens space, complex or
real projective space, is equivariantaly diffeomorphic to a linear action. We
show that a compact, simply connected Riemannian 4- or 5-manifold of
quasipositive curvature and maximal symmetry rank must be diffeomorphic to the
4-sphere, complex projective plane or the 5-sphere.
