In 1965, SS Chern posed the following question [7, p. 167](sometimes called Chern's
conjecture [12, p. 671]; see also [6]): Let M be a compact Riemannian manifold of positive
sectional curvature. Is it true that every abelian subgroup of 7i" i (M) is cyclic? Since 7i" i (M)
is finite, this is equivalent to saying that the cohomology ring H*(TTI, Z) is periodic (cf.[2]). In
this note we will point out that there exist infinitely many counterexamples by observing that
the normal homogeneous Aloff-Wallach space Niti (cf.[10]) and the Eschenburg space M^ i …

Let M be a closed oriented Riemannian manifold of dimension 5 with positive sectional
curvature. If M admits a π1-invariant isometric Tk (k= 2, 3), it has been shown by Fang and
Rong that M is homeomorphic to a spherical space form. In this short paper, we show that if
M admits a π1-invariant isometric T3-action, then π1 (M) is actually cyclic. Furthermore, we
show that if π1 (M) is not isomorphic to Z3 as well, then M is diffeomorphic to a lens space.