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Let $(M,g)$ be an $n-$dimensional complete open Riemannian manifold withnonnegative 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 thedistance function from a given point. If $(M,g)$ supports a second-orderSobolev inequality with a constant $C>0$ close to the optimal constant $K_0$ inthe second-order Sobolev inequality in $\mathbb R^n$, we show that a globalvolume non-collapsing property holds on $(M,g)$. The latter property togetherwith a Perelman-type construction established by Munn (J. Geom. Anal., 2010)provide several rigidity results in terms of the higher-order homotopy groupsof $(M,g)$. Furthermore, it turns out that $(M,g)$ supports the second-orderSobolev inequality with the constant $C=K_0$ if and only if $(M,g)$ isisometric to the Euclidean space $\mathbb R^n$.

Second-order Sobolev inequalities on a class of Riemannian manifolds with nonnegative Ricci curvature