In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold $(M^{ n},g)$ with scalar curvature $R_{ g}\geq 6$ admits a non-zero degree and $1$-Lipschitz map to $(\mathbb{ S}^{ 3}\times \mathbb{ T}^{ n-3},g_{ \mathbb{ S}^{ 3}}+g_{ \mathbb{ T}^{ n-3}})$, for $4\leq n\leq 7$, then $(M^{ n},g)$ is locally isometric to $\mathbb{ S}^{ 3}\times\mathbb{ T}^{ n-3}$. Similar results are established for noncompact cases as $(\mathbb{ S}^{ 3}\times \mathbb{ R}^{ n-3},g_{ \mathbb{ S}^{ 3}}+g_{ \mathbb{ R}^{ n-3}})$ being model spaces (see Theorem \ref{ noncompactrigidity1}, Theorem \ref{ noncompactrigidity2}, Theorem \ref{ noncompactrigidity3}, Theorem \ref{ noncompactrigidity4}). We observe that the results differ significantly when $n=4$ compared to $n\geq 5$. Our results imply that the $\epsilon$-gap length extremality of the standard $\mathbb{ S}^3$ is stable under the Riemannian product with $\mathbb{ R}^m$, $1\leq m\leq 4$ (see $D_{ 3}$. Question in Gromov's paper \cite{ Gromov2017}, p.153).
Categories: math.DG