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

## TL;DR

This paper establishes a differentiable sphere theorem for manifolds with a single cut point, showing that small radial curvature deviations imply diffeomorphism to the original manifold, extending classical results like the Cartan--Ambrose--Hicks theorem.

## Contribution

It introduces a weak differentiable sphere theorem for manifolds with a single cut point, connecting curvature closeness to diffeomorphism, and generalizes previous results including Cheeger's theorem.

## Key findings

- Manifolds with a single cut point are diffeomorphic under small radial curvature deviations.
- The result provides a weak version of the Cartan--Ambrose--Hicks theorem.
- It extends the Blaschke conjecture for spheres and includes exotic spheres in the scope.

## 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 $p$. 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.