Differentiable sphere theorems whose comparison spaces are standard spheres or exotic ones

Abstract

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 $p$.