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Put $\theta (r)=\mathrm{Vol}B(p,r)/({\omega }_n{r}^n)$. The Bishop-Gromov inequality says that $\theta$ is nonincreasing and at most one, while smoothness gives $\theta (r)\to 1$ as $r\downarrow 0$. The prescribed limit at infinity is also one. Hence $\theta (r)=1$ for every $r>0$.

Here is how equality gives the global Euclidean rigidity of maximal asymptotic volume ratio. Assume first $n\ge 2$. In geodesic polar coordinates centered at $p$, let $c(\xi )$ be the cut time along unit direction $\xi$, and $J(r,\xi )$ the positive radial volume density before that time. The comparison giving the Bishop-Gromov inequality says $J(r,\xi )\le r^{n-1}$, andNo cut time can be finite. If $c({\xi }_0)<t$, then the radial segment at time $t$ is not minimizing, so $d(p,{\exp }_p(t{\xi }_0))<t$. This strict inequality persists for nearby directions by continuity, and their cut times are also at most $t$. At any radius $R>t$, that open set of directions omits all radial volume from $t$ to $R$. Even the maximal density $r^{n-1}$ cannot then give Euclidean volume, contradicting $\theta (R)=1$. Thus $c(\xi )=\infty$ for all directions. Equality of all ball volumes and continuity now force $J(r,\xi )=r^{n-1}$ everywhere.

Let $A(r)X={\nabla }_X{\partial }_r$ on the distance sphere. This is the negative of the outward shape operator in the convention $S=-\nabla \nu$. Put $h=\mathrm{tr}A={\partial }_r\log J$. The traced radial Riccati equation for distance spheres isSince $h=(n-1)/r$, it givesBut $\mathrm{tr}(A^2)\ge h^2/(n-1)=(n-1)/r^2$ by the Cauchy-Schwarz inequality, and the Ricci term is nonnegative. Equality therefore holds in both, so $A=r^{-1}I$. Writing the polar metric as $d{r}^2+g_r$, its evolution is ${\partial }_r{g}_r=2g_r/r$. Its small-radius Euclidean limit then givesThe Riemannian exponential map is onto by the Hopf-Rinow theorem, has no conjugate points, and is injective because there is no finite cut time. Explicitly, two distinct minimizing radial geodesics reaching the same point would, after extending one past that point, create a length-minimizing broken curve with a genuine corner; smoothing the corner shortens it. That contradicts global minimization of every radial segment. Thus ${\exp }_p$ is a global diffeomorphism, and the displayed metric provesIn dimension one, a connected complete manifold without boundary is a line or a circle. The circle has limiting volume ratio zero, so the hypothesis leaves only the Euclidean line.