  Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold with
nonnegative Ricci curvature verifying $\rho\Delta_g \rho \geq n- 5\geq 0$,
where $\Delta_g$ is the Laplace-Beltrami operator on $(M,g)$ and $\rho$ is the
distance function from a given point. If $(M,g)$ supports a second-order
Sobolev inequality with a constant $C>0$ close to the optimal constant $K_0$ in
the second-order Sobolev inequality in $\mathbb R^n$, we show that a global
volume non-collapsing property holds on $(M,g)$. The latter property together
with a Perelman-type construction established by Munn (J. Geom. Anal., 2010)
provide several rigidity results in terms of the higher-order homotopy groups
of $(M,g)$. Furthermore, it turns out that $(M,g)$ supports the second-order
Sobolev inequality with the constant $C=K_0$ if and only if $(M,g)$ is
isometric to the Euclidean space $\mathbb R^n$.
