  We show that for an arbitrarily given closed Riemannian manifold $M$
admitting a point $p \in M$ with a single cut point, every closed Riemannian
manifold $N$ admitting a point $q \in N$ with a single cut point is
diffeomorphic to $M$ if the radial curvatures of $N$ at $q$ are sufficiently
close in the sense of $L^1$-norm to those of $M$ at $p$. Our result hence not
only produces a weak version of the Cartan--Ambrose--Hicks theorem in the case
where underlying manifolds admit a point with a single cut point, but also is a
kind of a weak version of the Blaschke conjecture for spheres proved by Berger.
In particular that result generalizes one of theorems in Cheeger's Ph.D. Thesis
in that case. Remark that every exotic sphere of dimension $> 4$ admits a
metric such that there is a point whose cut locus consists of a single point.
