Llarull type theorems on complete manifolds with positive scalar curvature

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≥ 6 admits a non-zero degree and 1-Lipschitz map to (𝕊^3×𝕋^n-3,g_𝕊^3+g_𝕋^n-3), for 4≤ n≤ 7, then (M^n,g) is locally isometric to 𝕊^3×𝕋^n-3. Similar results are established for noncompact cases as (𝕊^3×ℝ^n-3,g_𝕊^3+g_ℝ^n-3) being model spaces (see Theorem , Theorem , Theorem , Theorem ). We observe that the results differ significantly when n=4 compared to n≥ 5. Our results imply that the ϵ-gap length extremality of the standard 𝕊^3 is stable under the Riemannian product with ℝ^m, 1≤ m≤ 4 (see D_3. Question in Gromov's paper , p.153).