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 ∈ M with a single cut point, every closed Riemannian manifold N admitting a point q ∈ 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 L¹-norm to those of M at p.