  We show that an odd dimensional closed manifold with positive curvature
cannot contain an incompressible real projective plane in the sense that there
is no map of the projective plane into the manifold which is nontrivial on both
first and second homotopy groups. Another way to say this is that no element of
the fundamental group reverses the orientation of a class in the second
homotopy group. Further we show that for a manifold of dimension divisible by
four with positive curvature the fundamental group acts trivially on the second
homotopy group. The methods involve a careful study of the stability of minimal
two spheres in manifolds of positive curvature.
