## Riemannian manifolds with positive radial curvature.(English)Zbl 0804.53059

The authors prove the following results.
Theorem A: A complete noncompact Riemannian manifold $$M$$ of positive minimal radial curvature has exactly one end.
Theorem B: If the minimal radial curvature of $$M$$ is bounded below by 1 and if the volume of $$M$$ is greater than 3/4 $$\omega_ n$$, where $$\omega_ n$$ is the volume of the unit $$n$$-sphere $$S^ n$$, then $$M$$ is homeomorphic to $$S^ n$$.
Theorem C: If $$M$$ has non-negative minimal radial curvature with base point at $$0\in M$$ and satisfies $$\lim_{r\to\infty} {\text{vol }B(0,r)\over b_ 0(r)}> {1\over 2}$$, where $$b_ 0(r)$$ is the volume of the $$r$$-ball of a complete and simply-connected Riemannian $$n$$-manifold of constant zero curvature, then $$M$$ is diffeomorphic to $$\mathbb{R}^ n$$.
Theorem D: For given constants $$\kappa\in \mathbb{R}$$ and $$n\geq 2$$ there exists a constant $$\varepsilon_ 0= \varepsilon_ 0(n,\kappa)> 0$$ such that if $$M$$ has the properties $$\dim M= n$$, $$k_ M\geq -\kappa^ 2$$, $$k_ 0\min\geq 1$$ and $$\text{vol}(M)\geq \omega_ n- \varepsilon$$ for $$\varepsilon\in (0,\varepsilon_ 0)$$, where $$k_ M$$ is the sectional curvature of $$M$$, then there exists a diffeomorphism \hbox{$$f: M\to S^ n(1)$$} with the property that there exists a constant $$\tau= \tau(n,\kappa,\varepsilon)> 0$$ with $$\lim_{\varepsilon\to 0}\tau= 0$$ and $$e^{-\tau}\leq{\| df(\xi)\|\over \|\xi\|}\leq e^ \tau$$ for every non-zero tangent vector $$\xi$$ to $$M$$.

### MSC:

 53C20 Global Riemannian geometry, including pinching

### Keywords:

sphere; Euclidean space; minimal radial curvature