# Non-negatively curved manifolds and the volume of balls

Whether a complete non-compact non-flat Riemannian $$n$$-manifold $$M$$ with non-negative sectional curvature has Euclidean volume growth?

That is, whether there is a constant $$C>0$$ such that $$\mathrm{Vol}(B_x(r))\geq Cr^n$$ for all $$r>0$$ and $$x\in M$$? Here $$\mathrm{Vol}(B_x(r))$$ is the volume of the $$r$$-ball in $$M$$.

Since manifolds with non-negative Ricci curvature and Euclidean volume growth are studied a lot, I am curious about the non-negative sectional curvature case.

It is certainly not true that every complete nonflat open manifold of nonnegative curvature has Euclidean volume growth. Counterexamples are trivial to construct. Say, a capped cylinder. More generally any nonnegatively curved manifold $$M^n$$ with nontrivial soul has slower than Euclidean volume growth. Because its asymptotic cone at infinity has dimension strictly smaller than $$n$$ if the soul is not a point while manifolds with Euclidean volume growth have asymptotic cones of dimension $$n$$.
This also implies that any nonnegatively curved manifold with Euclidean volume growth is diffeomorphic to $$\mathbb R^n$$.
• if you are asking whether a nonnegatively curved manifold with Euclidean volume growth (meaning $vol B_r(p)\ge cr^n$) must be isometric to $\mathbb R^n$ that is certainly false. if you are asking something else please clarify. Apr 11 at 3:24
• Yes, there is my question. And whether a complete non-compact Alexandroff spaces (with n-dimensional Hausdorff dimension or n-dimensional topological dimension) with curvature bounded below by 0 and Euclidean volume growth is homeomorphic to $\mathbb{R} ^n$? Apr 11 at 3:31